REESE  LIBRARY 

i»i-   i  ii  i. 

UNIVERSITY  OF  CALIFORNIA. 


METHODS 


TEACHING  ARITHMETIC 


PRIMARY    SCHOOLS  * 


BY 

LARKIN   DUNTON,   LL.D. 

HEAD-MASTER  OF  THE  BOSTON  NORMAL  SCHOOL 


— 00^00 

"         OF  THE 

;tJNIVERSITTj 


SILVER,  BURDETT  &  CO.,  PUBLISHERS 
NEW  YORK  .  .  .  BOSTON  .  .  .  CHICAGO 
I  891 


COPYRIGHT,  1887, 
BY  LARKIN   DUNTON- 


TYPOGRAPHY  BY  J.  S.  GUSHING  &  Co.,  BOSTON. 
PRESSWORK  BY  BERWICK:  &  SMITH,  BOSTON. 


OF  THE 

(TJNITBRSITT, 
PREFACE. 


IT  has  long  seemed  to  the  author  that  there  was  needed 
in  this  country  a  manual  for  teachers,  which  should  set 
forth,  in  systematic  order  and  fully,  the  process  of  procedure 
on  the  part  of  the  teacher  in  developing  in  the  pupils  ideas 
of  numbers  and  their  relations,  ideas  of  numerical  processes, 
and  ideas  of  the  signs  by  which  both  are  represented.  The 
present  work  is  an  attempt  to  supply  that  need  in  the  case 
of  teachers  of  primary  schools. 

For  illustrations  and  method  of  treatment  the  author  is 
mainly  indebted  to  the  work  of  A.  Bohme,  formerly  a  nor- 
mal school  teacher  in  Berlin,  Anleitung  zum  Unterricht  im 
Rechnen.  But  Bohme  is  not  responsible  for  any  of  the 
statements  made  in  this  work ;  because  such  omissions, 
additions,  and  other  changes  have  been  made  as  seemed 
desirable  in  order  to  adapt  the  work  to  the  needs  of  Ameri- 
can schools.  While  portions  of  this  work  are  free  transla- 
tions from  Bohme,  others  are  independent  discussions. 

The  scope  of  the  work  and  the  arrangement  of  topics 
can  be  seen  by  a  glance  at  the  table  of  contents. 

While  frequent  reference  is  made  to  Arithmetic  Charts 
by  the  same  author,  the  methods  here  described  are  not 


PREFACE. 


dependent  upon  any  particular  kind  of  apparatus,  but  are 
equally  applicable,  whatever  may  be  the  apparatus  in  use. 

If,  by  the  introduction  of  this  manual,  and  of  his  Arith- 
metic Charts,  the  author  shall  have  made  the  teaching  of 
arithmetic  easier  for  the  teacher,  and  more  educational  for 
the  pupil,  his  purpose  in  preparing  them  will  have  1 
accomplished. 


BOSTON,  Sept.  15,  1888. 


CONTENTS. 


CHAPTER  I. 

PAGE 

NUMBERS  FROM  ONE  TO  TEN 7 

Counting 7 

Apparatus  for  illustration 8 

Use  of  apparatus  in  counting 1 1 

Slate  exercises  in  counting 14 

Number  pictures 17 

Arithmetic  charts 22 

Separating  numbers  into  two  parts 24 

Learning  the  use  of  figures 37 

Written  exercises  in  addition 38 

Written  exercises  in  subtraction 40 

Combined  addition  and  subtraction 42 

CHAPTER  II. 

NUMBERS  FROM  ONE  TO  TWENTY 44 

Counting 44 

Separating  numbers  into  two  parts 47 

Teaching  numbers  from  1 1  to  20 53 

CHAPTER  HI. 

NUMBERS  FROM  ONE  TO  ONE  HUNDRED 61 

Counting 61 

Reading  and  writing  numbers 65 

Addition 67 


6  CONTENTS. 

PAGE 

Subtraction 75 

Addition  and  subtraction      . 78 

Multiplication 82 

Teaching  the  multiplication  table 83 

Applying  the  table  to  written  work 90 

Constructing  the  table 94 

Preparation  for  division 97 

Division 99 

Dividing  by  two 101 

Dividing  by  three TIO 

Dividing  by  other  numbers  to  ten 113 

Practice  work 1 14 

CHAPTER  IV. 

NUMBERS  FROM  ONE  TO  ONE  THOUSAND 117 

Counting  and  writing 117 

Addition 1 19 . 

Subtraction 122 

Multiplication 127 

Division 132 

Written  and  mental  arithmetic 137 

CHAPTER  V. 

HIGHER  NUMBERS 141 

Numeration 141 

Addition 147 

Subtraction 149 

Multiplication 152 

Division 157 


fes. 

Of  THE  \ 

UNIVERSITY) 


TEACHING    ARITHMETIC. 

CHAPTER   I. 

NUMBERS   FROM   ONE   TO   TEN. 

1.   COUNTING. 

INSTRUCTION  in  arithmetic  should  begin  with  count- 
ing, since  this  is  the  foundation  of  all  arithmetical 
operations. 

When  children  enter  school,  they  can  usually  count 
a  little,  that  is,  they  can  distinguish  a  few  numbers 
of  similar  things  by  the  appropriate  words.  But  it 
frequently  happens  that  the  children  know  words 
which  stand  for  numbers,  merely  as  a  succession  of 
sounds,  without  knowing  what  number  of  things  one 
word  or  another  signifies.  The  teacher  should  be 
careful  not  to  mistake  this  mechanical  knowledge  of 
words  for  a  real  knowledge  of  numbers,  or  for  the 
ability  to  count.  In  order  to  give  the  child  a  definite 
idea  of  the  meaning  of  the  number  words  which  are 
already  partly  known  to  him,  we  should  direct  his 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 

attention  to  the  objects  around  him,  and  teach  him 
the  words  which  express  their  numbers.  .  Through 
conversation  we  may  lead  him  to  observe  that  there 
are  in  the  room  one  door  and  another  door,  and  thus 
to  comprehend  the  expression  that  one  door  and  one 
door  are  two  doors.  When  the  meaning  of  the  word 
two  is  thus  made  clear,  it  may  be  applied  to  different 
objects  in  the  room,  as  to  two  hands,  two  feet,  two 
eyes,  two  ears,  two  windows,  etc. 

In  this  way  we  may  lead  the  child  to  the  compre- 
hension and  application  of  the  words  one,  two,  three, 
four,  etc.,  to  ten. 

We  should  thus  use  concrete  objects  in  making  the 
child  acquainted  with  numbers  and  the  naming  of 
numbers.  Since  clear  and  correct  ideas  are  based 
upon  perception  alone,  it  is  necessary  to  connect  the 
exercises  in  number  in  the  lower  grades  continually 
with  concrete  objects.  In  order  to  reach  the  proper 
result  in  the  shortest  time,  it  is  desirable  to  have 
some  simple  apparatus,  by  means  of  which  all  the 
necessary  observations  can  readily  be  made. 

2.   APPARATUS  FOR   ILLUSTRATION. 

Festal ozzi  furnished  such  an  apparatus  in  his  table 
of  units.  This  consisted  of  a  table  containing  ten 
rows  of  rectangles  with  ten  rectangles  in  each  row. 
In  each  rectangle  in  the  first  row  was  one  line,  in 
each  rectangle  of  the  second  row  were  two  lines,  and 


APPARATUS  FOR  ILLUSTRATION.  9 

so  on  to  the  tenth  row,  in  each  rectangle  of  which 
were  ten  lines. 

Other  machines  for  illustrating  number-teaching 
have  been  devised,  involving  the  same  principle,  and 
recognizing  fully  the  necessity  of  observation,  which 
have  been  improvements  upon  Pestalozzi's.  One  of 
these  consists  of  a  black  wooden  board,  about  twenty 
inches  long  and  twenty  inches  wide,  in  which  are 
bored  a  hundred  holes,  in  ten  rows  of  ten  each,  so 
arranged  that  there  are  ten  horizontal  and  ten  ver- 
tical rows.  In  addition  to  this  board  are  one  hun- 
dred buttons  of  white  wood  or  bone,  with  stems  that 
can  be  stuck  in  the  holes.  This  has  several  advan- 
tages over  the  fixed  table  of  Pestalozzi : 

i.    It  allows  each  exercise  to  be  observed  alone; 

2.  The    children    can    see    the    numbers    produced ; 

3.  They    can    themselves    perform    the     exercises ; 

4.  The  things  to  be  observed  are  objects,  and  con- 
sequently   much    better    than    any   signs ;     5.   The 
number  pictures  (to  be  explained  hereafter)  can  be 
formed.     Besides,  it  is  very  handy,  and  the  exercises 
to  be  shown  can  easily  be  observed  by  the  children 
who  sit  in  the  farther  part  of  the  room. 

Where  such  an  apparatus  is  wanting,  the  teacher 
may  use  for  the  same  purpose  wooden  pegs,  bits  of 
pasteboard,  cubes,  and  the  like.  Mothers  and  nurses 
might  do  much  to  prepare  the  children  for  instruc- 
tion and  education  in  numbers,  would  they  take  occa- 
sion to  have  them  count  and  compare,  while  at  play 


10  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

with  their  little  plates,  soldiers,  building  blocks,  etc. 
It  is  possible,  even  while  the  sole  purpose  is  play,  so 
to  direct  them  as  to  do  much  to  prepare  them  for 
instruction  and  for  the  development  of  their  powers. 

Another  piece  of  apparatus  is  the  numeral  frame, 
one  form  of  which  is  much  used  in  this  country. 
The  best  kind  is  composed  of  a  wooden  frame  about 
four  feet  long  and  two  feet  wide,  in  which,  running 
horizontally  from  end  to  end,  are  fastened  ten  brass 
or  steel  rods ;  on  each  of  these  rods  are  ten  easily 
moved  wooden  balls  about  an  inch  and  a  half  in 
diameter.  The  whole  is  supported  at  a  convenient 
height  by  means  of  upright  standards  attached  to 
bars  running  crosswise  at  the  bottom.  It  is  well  to 
have  the  balls  painted  different  colors,  say  three  red 
ones  at  the  left  on  each  wire,  then  three  yellow  ones, 
and  four  green  ones  at  the  right ;  or,  two  and  two, 
black,  red,  yellow,  green,  and  white.  One-half  the 
frame  should  be  covered  with  a  board,  so  as  to  con- 
ceal all  the  balls  that  are  not  used  in  any  example. 
This  apparatus  has  some  special  advantages.  It  can 
be  made  to  present  many  examples  very  readily,  the 
balls  can  be  seen  across  the  room,  and,  by  means  of 
the  different  colors,  the  number  of  balls  in  sight  on 
any  wire  can  be  readily  determined,  even  by  the 
farthest  children. 

The  two  pieces  of  apparatus  just  described  may  be 
united  in  one  by  boring  holes  for  the  buttons  in  the 
board  used  for  a  screen  for  the  balls,  as  shown  in  the 


USE    OF  APPARATUS  IN  COUNTING. 


II 


cut.  This  board  may  be  made  to  perform  still  an- 
other office  by  ruling  vertical  and  horizontal  lines 
across  it  through  the  rows  of  holes,  namely,  the  form- 
ing of  number  pictures,  to  be  explained  hereafter. 


There  are  other  ingeniously  constructed  machines 
for  teaching  the  first  steps  in  number ;  but  generally 
the  simplest  form  is  the  best.  Perhaps  for  general 
use  the  numeral  frame  that  I  have  just  described  is 
the  most  desirable. 


3.   USE  OF  APPARATUS  IN  COUNTING. 

However  necessary  real  observation  may  be  for  the 
first  steps  in  arithmetical  instruction,  and  even  in 
exercises  introduced  later,  yet,  ultimately,  observa- 
tion must  be  replaced  by  ideas ;  ideas  in  the  mind 


12         ARITHMETIC  IN  PRIMARY  SCHOOLS. 

must  take  the  place  of  objects  without.  The  child 
learns  to  walk,  at  first,  with  help,  then  independently ; 
he  learns  to  dispense  with  assistance  gradually.  So 
it  is  with  the  use  of  observation  in  arithmetical 
instruction. 

It  has  already  been  explained  how  a  child  may  be 
made  to  gain  clear  ideas  of  the  fundamental  numbers, 
that  is,  the  numbers  from  one  to  ten.  But  the  work 
thus  begun  may  be  completed  and  the  ideas  impressed 
upon  the  memory  by  means  of  the  numeral  frame, 
in  the  following  manner ;  or,  indeed,  the  very  first 
instruction  may  be  so  given  : 

Move  out  one  ball  on  the  upper  wire,  and  ask, 
"  How  many  balls  is  that  ?  "  The  answer  should  be 
given  in  a  complete  sentence,  thus,  "That  is  one 
ball."  Move  out  two  balls  on  the  second  wire,  and 
to  the  question,  "  How  many  balls  are  there  ?  "  should 
the  answer  follow,  "There  are  two  balls."  But  if 
the  child  does  not  know  the  word  two,  the  sentence 
must  be  given  first  by  the  teacher. 

In  the  same  way  show  the  class  all  the  numbers  of 
balls  from  one  to  ten,  and  teach  the  sentences,  in 
connection  with  the  observation,  "  There  is  one  ball," 
"There  are  two  balls,"  and  so  on  to,  "There  are  ten 
balls."  Let  these  sentences  be  clearly  and  distinctly 
pronounced,  now  by  individuals,  and  now  by  the 
class  in  concert,  while  the  teacher  points  successively 
to  the  different  groups  of  balls.  Finally,  let  the 
children  use  the  pointer. 


USE    OF  APPARATUS  IN  COUNTING.  13 

When  these  sentences  have  been  learned  as  the 
expressions  of  the  several  facts,  let  them  be  abbre 
viated,  thus  : 

1.  One  ball,  two  balls,  etc.,  to  ten  balls. 

2.  One,  two,  three,  etc.,  to  ten. 

Next,  teach  the  following  sentences  in  order,  in 
connection  with  the  use  of  the  balls  : 

1.  After 'one  comes  two,  after  two   comes  three, 
etc.,  to  ten. 

2.  One  and  one  are  two,  two  and  one  are  three, 
etc.,  to  ten. 

Although  the  last  two  sentences  are  not  exactly 
counting,  yet  they  are  in  place  here,  for  they  result 
immediately  from  the  facts  learned  in  counting. 

Up  to  this  point  we  have  had  the  children  speak 
the  sentences  in  the  order  of  the  numbers,  so  as  to 
accustom  them  to  the  counting  of  objects,  and  to  the 
use  of  number  words  in  their  natural  order ;  because 
counting  is  the  foundation  of  all  arithmetical  opera- 
tions. Now,  however,  exercises  may  be  given  out  of 
their  order,  so  that  the  pupils  may  be  led  to  count 
any  number  of  objects,  as  fingers,  windows,  etc.,  or 
to  tell  their  number. 

When  the  children  have  become  proficient  in  desig- 
nating any  number  of  objects  up  to  ten  by  the  appro- 
priate word,  in  telling  what  number  comes  after  each 
number,  and  how  many  each  number  becomes  when 
increased  by  one,  they,  may  then  be  taught  to  count 


14  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

backwards  from  ten  to  one.      The   proper  order  of 
the  steps  may  be  indicated  as  follows  : 

1.  There  are  ten  balls,  there  are  nine  balls,  etc.,  to 
one. 

2.  Ten  balls,  nine  balls,  etc.,  to  one. 

3.  Ten,  nine,  eight,  etc.,  to  one. 

4.  Before  ten  comes  nine,  before  nine  comes  eight, 
etc.,  to  one. 

5.  One  from  ten  leaves  nine,  one  from  nine  leaves 
eight,  etc.,  to  one. 

6.  Ten  less  one  is  nine,  nine  less  one  is  eight,  etc., 
to  one. 

The  method  is  the  same  as  in  counting  from  one 
to  ten.  The  numbers  and  their  relations  are  to  be 
suggested  by  the  observation  of  the  balls,  as  they 
are  presented  by  the  teacher. 

4.    SLATE  EXERCISES  IN  COUNTING. 

Many  classes  of  beginners  are  so  constituted  that 
the  pupils  are  not  all  of  the  same  degree  of  advance- 
ment, so  that  they  cannot  properly  be  taught  all 
together.  When  this  is  the  case,  we  have  little  need 
to  consider  the  question  of  slate  exercises  ;  for  these 
are,  at  this  stage  of  the  work,  merely  makeshifts  ; 
and,  however  closely  they  may  be  connected  with 
the  objects  to  be  observed,  the  real  teaching  of  the 
numbers  cannot  be  dispensed  with.  But  when  the 


SLATE   EXERCISES  IN  COUNTING.  15 

class  contains  two  or  more  divisions,  then  the  teacher 
must  provide  suitable  exercises  to  occupy  the  rest  of 
the  children  while  he  is  engaged  in  teaching  one 
division.  For  this  purpose  he  must  make  use  of  the 
blackboard.  Of  course,  exercises  in  writing  furnish 
abundant  means  for  occupation,  especially  where 
reading  and  writing  are  taught  together ;  but  exer- 
cises should  be  devised  which  satisfy  the  aim  of  arith- 
metical instruction.  All  exercises  designed  simply 
to  keep  the  children  busy  are  an  abomination.  When 
slate  exercises  are  introduced  as  a  means  of  teaching 
counting,  they  should  be  closely  connected  with  the 
objects  to  be  observed  by  the  children;  •so  that  the 
work  done  by  them  will  be  a  means  of  fixing  in  their 
minds  the  ideas  gained  by  the  observation  of  the 
objects. 

The  children  should  not  at  this  stage  be  made 
acquainted  with  figures,  for  they  are  not  yet  able  to 
represent  in  their  minds,  by  means  of  figures,  what 
the  figures  signify,  because  figures  are  purely  arbi- 
trary signs,  and  not  pictures  of  numbers,  in  which 
the  children  can  again  find  the  units  which  they 
signify.  Written  exercises  in  number  must,  there- 
fore, at  first,  consist  of  representations  of  numbers 
of  units. 

To  prepare  the  children  for  these  exercises,  the 
teacher  should  write  upon  the  board  the  following,  or 
similar  groups  of  marks,  while  the  children  observe 
and  count.  The  board  upon  which  these  groups  are 


i6 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


written  should  be  ruled  in  squares  of  convenient  size, 
so  that  each  line,  star,  etc.,  will  occupy  one  square. 


o 

o  o 
o  o  o 
o  o  o  o 
o  o  o  o  o 
o  o  o  o  o  o 
o  o  o  o  o  o  o 
oooooooo 

I  OOOOOOOOO 

II  OOOOOOOOOO 

II       OOOOOOOOOO 
I         OOOOOOOOO 

oooooooo 
o  o  o  o  o  o  o 
o  o  o  o  o  o 
o  o  o  o  o 
o  o  o  o 
o  o  o 
o  o 


These  groups  may  be  formed  of  other  figures,  at 
the  pleasure  of  the  teacher ;  for  example, 

DO  +    X    T    I. 

The  upper  row  of  groups  represents  counting  for- 
wards, and  each  row  of  marks  is  one  more  than  the 
row  above ;  while  the  lower  row  of  groups  represents 
counting  backwards,  and  each  row  of  marks  is  one 
less  than  the  row  above. 

The  different  lines,  figures,  etc.,  which  the  teacher 
can  increase  at  his  pleasure,  serve  both  for  a  change, 
and  for  practice  in  writing  and  drawing.  The  slates 
should  be  ruled  in  squares  corresponding  with  the 


NUMBER  PICTURES.  \J 

ruling  of  the  board ;  which  may  be  done  by  scratch- 
ing the  slates  lightly.  This  marking  of  the  slates 
assists  in  establishing  the  habit  of  doing  all  work  on 
the  slate  in  an  orderly  manner.  It  is  of  much  use  a 
little  later  in  fixing  the  habit  of  writing  figures  of 
uniform  size,  and  in  vertical  and  horizontal  lines. 

When  the  children  have  seen  the  groups  formed, 
have  counted  the  marks,  compared  and  named  them, 
so  that  their  numbers  are  all  well  known,  they  may 
be  required  to  copy  them  ;  and  the  closer  they  are 
required  to  follow  the  copy,  the  better,  not  only  for 
the  training  of  the  eye  and  hand,  but  also  for  the 
arithmetic  itself ;  for,  as  exactness  in  the  order  light- 
ens the  work  of  computing  the  numbers,  so  it  makes 
clearer  the  knowledge  of  the  relation  of  one  number 
to  another.  Moreover,  exactness  here  is  of  great 
moral  value,  inasmuch  as  it  trains  to  habits  of  order 
and  neatness. 

Chart  I.  will  now  be  found  useful  for  review  in 
counting  and  for  copying. 

5.    NUMBER  PICTURES. 

Through  the  exercises  already  explained,  the  chil- 
dren may  gain  ideas  of  all  the  fundamental  numbers, 
—  that  is,  all  the  numbers  from  one  to  ten,  —  in  their 
unity.  The  eye,  however,  is  not  in  condition  to  see 
a  large  number  of  units  lying  side  by  side,  or  one 
above  another,  and  grasp  the  units  as  a  number. 


i8 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


The  children,  for  the  most  part,  have  merely  the 
power  to  count, — a  power  which  is  to  be  regarded 
only  as  the  foundation.  If  a  child  is  to  obtain  a  total 
impression  of  a  number  at  once,  the  number  must  be 
in  the  form  of  a  definite  picture,  in  which  he  discov- 
ers the  number  at  a  glance,  and  grasps  it  immedi- 
ately in  all  its  units.  Such  number  pictures  are  pre- 
sented at  the  bottom  of  Chart  I. 


CHART    I. 

FOR  REVIEW  IN  COUNTING. 


•  € 

• 


NUMBER   PICTURES. 


NUMBER  PICTURES.  19 

These  rectangles  with  the  enclosed  dots  should  be 
put  upon  the  board,  one  after  another,  and  when  they 
have  been  observed  and  the  dots  counted,  they  should 
be  carefully  copied  on  the  slates,  in  order  to  impress 
them  upon  the  eye  and  memory. 

In  regard  to  one,  two,  and  three,  there  is  nothing 
of  importance  to  be  said. 

In  four  we  see  two  points  above  and  two  points 
below ;  or  two  points  at  the  right  and  two  points  at 
the  left.  Attention  may  be  directed  to  the  form  of 
the  picture  by  questions  :  "  What  do  you  see  in 
four?"  "Two  points  above  and  two  points  below." 
"What  else?"  "Two  points  at  the  right  and  two 
points  at  the  left."  But  it  will  be  taken  for  granted 
generally  in  this  book,  that  the  teacher  knows  how 
to  develop  the  points  of  a  lesson  by  proper  questions, 
when  they  are  suggested. 

As  the  children  copy,  the  arrangement  of  the 
dots  makes  clear  to  them  the  thoughts  expressed 
by  the  following  sentences,  which  may  be  devel- 
oped by  the  proper  questioning  on  the  part  of  the 
teacher : 

1.  From  four  we  can  make  two  twos. 

2.  Two  and  two  are  four. 

3.  Four  less  two  is  two. 

4.  Two  times  two  are  four. 

5.  The  half  of  four  is  two. 

6.  There  are  two  twos  in  four. 


20  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

In  order  to  give  further  practice  in  the  use  of  these 
sentences,  refer  to  objects  in  which  the  number  four 
appears ;  e.g.,  a  wagon  has  four  wheels,  two  before 
and  two  behind ;  the  cat,  dog,  mouse,  etc.,  have  each 
four  feet ;  the  table  has  four  legs,  etc. 

Occasionally  should  practical  problems  be  given  : 
George  has  two  cents,  and  gets  two  more  ;  how  many 
has  he  now  ?  And  so  of  the  other  relations  of  the 
numbers.  But  the  use  of  these  problems  should  not 
be  carried  too  far,  otherwise  the  arithmetical  instruc- 
tion lacks  brevity  and  definiteness.  I  shall  not  intro- 
duce these  problems  often,  because  the  live  teacher 
can  easily  invent  enough  to  fit  the  work  upon  the 
numbers,  as  they  are  studied,  one  after  another;  or, 
better  yet,  can  find  some  good  books  of  problems. 

The  number  Jive  may  be  produced  from  four  by 
putting  a  dot  in  the  midst  of  the  four. 

From  jfa^  we  can  make  four  and  one  ;  it  follows  that 

a.  4  and  I  are  5.  c.   5  less  I  is  4. 

b.  i  and  4  are  5.  d.   5  less  4  is  i. 

Six  consists  of  two  threes  ;  it  follows  that 

a.  3  and  3  are  6.  c.   2  times  3  are  6. 

b.  6  less  3  is  3.  d.   Half  of  6  is  3. 

e.  There  are  2  threes  in  6. 

Furthermore, 

f.  2  and  2  and  2  are  6.     k.  The  third  of  6  is  2. 

g.  3  times  2  are  6.  i.  There  are  3  twos  in  6. 

j.   2  and  2  are  4 ;  4  and  2  are  6. 


NUMBER  PICTURES.  21 

Seven  consists  of  six  and  one,  the  one  being  in  the 
middle ;  hence, 

a.  6  and  i  are  7.  c.  7  less  i  is  6. 

b.  i  and  6  are  7.  d.  7  less  6  is  i. 

Eight  consists  of  two  fours  ;  therefore 

a.  4  and  4  are  8.  c.  2  times  4  are  8. 

b.  8  less  4  is  4.  ^/.   Half  of  8  is  4. 

e.  There  are  2  fours  in  8. 

It  is  further  obvious  from  the  picture,  that  2  and  2 
and  2  and  2  are  8  ;  or  4  times  2  are  8  ;  the  fourth  of 
8  is  2  ;  and  there  are  4  twos  in  8. 

If  the  children  are  old  enough  and  advanced  enough 
to  make  it  easy  for  them  to  comprehend,  the  follow- 
ing facts  may  be  taught : 

I  of  8  is  2,  If  i  apple  costs  2  cents, 

f  of  8  are  4,  2  apples  cost  4  cents, 

f  of  8  are  6,  3  apples  cost  6  cents, 

|  of  8  are  8.  4  apples  cost  8  cents. 

\  of  8  away,  6  is  left,  If  i  orange  costs  8  cents, 

|  of  8  away,  4  is  left,  |  orange  costs  4  cents, 

f  of  8  away,  2  is  left,  \  orange  costs  2  cents, 

|  of  8  away,  o  is  left.  f  orange  cost  6  cents. 

Nine  consists  of  three  threes  ;  therefore 

a.  3  and  3  and  3  are  9.     d.  9  less  6  is  3. 

b.  3  times  3  are  9.  e.  6  and  3  are  9. 

c.  9  less  3  is  6.  f.  A  third  of  9  is  3,  etc. 


22  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

Ten  consists  of  two  fives  ;  therefore, 

a.  5  and  5  are  10.  c.    10  less  5  is  5. 

b.  2  times  5  are  10.          d.  Half  of  10  is  5. 

e.  There  are  2  fives  in  10. 

While  the  teacher  is  instructing  the  children  in 
these  numbers,  he  must  be  careful,  both  in  the  oral 
and  written  work,  to  make  them  able  to  name  the 
number  pictures  as  soon  as  they  are  seen,  and  also  to 
construct  them  on  the  numeral  frame,  or  draw  them 
on  the  board. 

An  excellent  practice  in  comparing  numbers  grows 
out  of  forming  one  number  from  another.  For  exam- 
ple, put  the  number  picture  for  five  on  the  board, 
then  ask,  What  must  be  done  in  order  to  make  a 
seven  ?  Must  something  be  added,  or  taken  away  ? 
How  many  must  be  added  ?  Where  must  the  dots 
be  put  ?  Again,  How  can  seven  be  made  from  nine  ? 
etc.  In  all  such  cases,  one  number  picture  is  to  be 
changed  to  another  by  either  addition  or  subtraction 
of  the  proper  dots.  Exercises  of  this  kind  are  very 
useful ;  they  exercise  the  children  in  the  comparison 
of  numbers,  and  prepare  them  for  the  division  of 
numbers,  which  is  about  to  be  explained  in  detail. 

6.    ARITHMETIC  CHARTS. 

In  addition  to  the  apparatus  for  developing  ideas 
of  number,  which  has  been  already  described,  a  few 


ARITHMETIC   CHARTS.  23 

arithmetic  charts  will  be  found  very  helpful  for 
reviews  at  every  stage  of  elementary  instruction  in 
arithmetic.  The  author  of  this  book  has  arranged  a 
series  of  thirteen  such  charts,  which  are  published 
by  Silver,  Burdett  &  Co.  Miniature  copies  of  them 
will  be  printed  in  this  book,  as  they  are  needed  for 
illustration  ;  and  they  will  be  referred  to  simply  by 
their  numbers.  Where  they  are  not  furnished  to 
schools,  teachers  can  put  them  on  the  blackboard,  or 
on  large  sheets  of  paper,  and  thus  save  themselves 
much  labor. 

A  word  in  regard  to  the  use  of  the  charts.  The 
children  should  see  each  number  and  each  exercise 
produced ;  that  is,  each  illustration  of  a  number,  or 
of  a  numerical  operation,  should  be  made  by  the 
teacher,  either  upon  the  numeral  frame,  or  upon  the 
board,  or  with  objects,  just  when  it  is  needed  to 
make  the  truth  clear  to  the  class.  Hence  all  ready- 
made  charts,  or  other  illustrations,  are  to  be  used 
later,  after  this  preliminary,  but  fundamental  work 
has  been  done.  They  are  to  serve  as  a  means  for 
review  and  practice  in  what  has  already  been  made 
clear  to  the  understanding. 

The  charts  representing  matter  for  observation  are 
to  be  read  forwards,  backwards,  vertically,  and  hori- 
zontally. The  special  use  to  be  made  of  the  differ- 
ent charts  will  be  explained  as  they  are  introduced. 
In  general,  they  are  designed  to  lighten  the  labor 
of  the  teacher,  while  making  the  instruction  more 
thorough  and  systematic. 


24  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

7.    SEPARATING  NUMBERS  INTO  Two. PARTS. 

Upon  a  pupil's  facility  in  the  use  of  numbers  below 
ten  depends  his  progress  in  mastering  numbers 
above  ten.  The  greater  his  facility  in  the  use  of 
small  numbers,  if  it  is  founded  upon  clear  under- 
standing, the  surer  and  more  rapid  will  be  his  prog- 
ress in  larger  numbers.  In  order  to  attain  this 
facility  depending  upon  understanding,  we  must  have 
the  numbers  regarded  from  as  many  sides  as  possi- 
ble ;  this  comes  from  the  division,  or  separation,  of 
the  numbers  into  their  component  parts.  From  this 
division  we  obtain  results  for  all  the  different  funda- 
mental operations  in  arithmetic,  which  are  the  more 
easily  committed  to  memory,  because  they  are  all 
grounded  upon  a  single  result,  namely,  that  of  divi- 
sion. The  results,  however,  which  are  obtained  from 
this  division,  must,  by  no  means,  be  learned  by  heart, 
as  one  commits  to  memory  vocabularies  or  verses ; 
they  must  become  things  of  the  memory  through  an 
unlimited  amount  of  reckoning,  —  through  practice. 
It  is  sufficient,  at  first,  that  a  child,  if  he  is  to  unite, 
for  example,  five  and  three  into  a  single  number,  adds 
first  one  to  five,  then  another,  and  still  another,  even 
if  he  represents  the  process  to  his  senses  by  means 
of  marks,  fingers,  etc.  ;  yet  continued  practice  must 
bring  him  to  the  point  where  the  union  of  three  and 
five  in  eight  is  a  simple  conception,  a  thing  of  the 
memory.  If  the  child  constantly  perceives  the  three 


SEPARATING  NUMBERS  INTO   TWO  PARTS.       2$ 

units  in  three,  he  will,  in  time,  be  able  to  unite  three 
to  five  at  once.  We  shall  be  able  to  bring  him  to 
this  state  of  mind  the  more  easily  if  we  show  him 
that  eight  consists  of  a  five  and  a  three.  When, 
however,  we  have  shown  him  this,  he  will  be  able, 
from  the  single  observation,  to  understand  the  fol- 
lowing four  sentences  : 

a.  Three  and  five  make  eight. 

b.  Five  and  three  make  eight. 

c.  Three  from  eight  leaves  five. 

d.  Five  from  eight  leaves  three. 

If  now,  we  use  these  sentences  as  the  expressions 
for  the  truths  which  constantly  appear  before  the 
eyes  of  the  child,  the  results  will  finally  become 
impressed  upon  his  memory.  This  result  will,  of 
course,  be  reached  in  the  case  of  some  children 
quicker  than  with  others. 

Out  of  the  above  division  we  obtain  two  results  in 
addition  and  two  in  subtraction.  Another  example 
will  show  that  in  a  single  division  all  the  four  funda- 
mental operations  of  arithmetic  may  be  illustrated. 
Out  of  eight  we  can  make  two  fours.  It  follows  that 

a.  4  and  4  are  8.  c.   8  less  4  is  4. 

b.  2  times  4  are  8.          d.  The  half  of  8  is  4. 

e.  There  are  2  fours  in  8. 

In  the  first  stages  of  arithmetical  work,  where  the 
numbers  are  small,  and  the  results  to  be  gained 


"26 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


through  division  are  correspondingly  few,  the  direct 
observation  may  result  in  clear  mental  pictures,  or 
ideas.  Here  the  connection  between  the  various 
ground  operations  of  arithmetic  is  so  obvious  from 
the  observation,  that  it  seems  unnecessary  to  sepa- 
rate the  treatment  of  the  different  operations.  Hence, 
in  this  and  the  following  stage,  that  is,  in  the  treat- 
ment of  numbers  from  one  to  ten,  and  from  ten  to 
twenty,  the  four  fundamental  operations  may  be 
united.  All  the  different  results  are  obtained  because 
every  number  below  ten  is  divided  into  every  two  parts 
of  which  it  is  composed,  in  the  way  shown  in  Chart  II. 

CHART   II. 

UNITING  AND  SEPARATING  NUMBERS  FROM  Two  TO  TEN. 


• 


>  •  • 


•   c 


I  will  now  give  an  explanation  of  Charts  I.  and  II. 
and  their  use. 

The  upper  part  of  Chart  I.  is  for  review  work  in 
counting.  The  dots  may  be  counted  under  the  direc- 


SEPARATING  NUMBERS  INTO   TWO  PARTS.        2/ 

tion  of  the  teacher ;  and  then  they  may  be  copied  on 
the  slates.  During  the  work  of  copying,  the  children 
should  always  count  the  dots  as  they  make  them. 

The  lower  part  of  Chart  I.  contains  the  number 
pictures  from  one  to  ten.  These  pictures  are  designed 
to  furnish  a  means  of  impressing  the  ideas  of  the 
fundamental  numbers,  —  that  is,  the  numbers  from 
one  to  ten,  —  upon  the  mind  in  such  a  way  that  they 
may  reappear  in  the  imagination  of  the  -  pupil  when- 
ever needed. 

These  pictures  should  not  be  used  as  the  sole 
means  of  developing  ideas  of  numbers,  but  rather  as 
a  means  of  thorough  review  and  impression.  Figures 
should  not  be  taught  in  connection  with  these  pict- 
ures. The  whole  attention  of  the  <pupil  should  be 
given  to  the  numbers  and  their  production  ;  mere 
figures  will  be  prominent  enough  in  his  work  by  and 
by,  however  much  pains  may  be  taken  to  avoid  it. 

Chart  II.  is  to  show  the  parts  of  numbers  to  ten. 
Each  number  picture  is  to  be  formed  at  first  by 
the  teacher  on  the  numeral  frame,  the  blackboard,  or 
other  apparatus  ;  so  that  the  attention  of  the  children 
can  be  directed  to  only  one  number.  All  the  special 
facts  in  regard  to  the  composition  of  the  number,  and 
the  relation  of  its  parts,  are  to  be  developed  by  proper 
questions,  and  by  pointing  to  the  parts  to  be  seen. 

As  fast  as  the  number  pictures  have  been  treated 
in  this  way  they  may  be  copied  from  the  chart,  and 
thus  much  labor  on  the  part  of  the  teacher  may  be 


28 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


saved.  After,  for  example,  the  number  two  has  been 
treated  as  indicated  below,  the  first  rectangle  may  be 
copied  by  the  children.  By  this  means  the  truths 
will  be  further  impressed  upon  the  mind..  The 
teacher,  however,  should  be  sure  that  the  children 
connect  the  proper  name  of  the  number,  and  the 
names  of  the  parts,  with  what  they  write ;  so  that 
numbers  and  names  will  become  thoroughly  asso- 
ciated in  their  minds. 

THE   NUMBER  TWO. 


This  is  the  first  rectangle  on  Chart  II.  It  shows 
that  the  number  tzvo  can  be  divided  into  two  units  ; 
hence  the  truth  of  the  following  sentences  : 

a.    i  and  I  are  2.  c.   2  less  I  is  I. 


b.  2  times  i  are  2. 


d.  1  of  2  is  I. 


e.    i  in  2  two  times. 


THE   NUMBER   THREE. 


From  three  we  can  make  a  two  and  a  one ;    it 
follows  that 

a.  2  and  i  are  3.  c.  3  less  i  is  2. 

b.  i  and  2  are  3.  d.  3  less  2  is  i. 


SEPARATING  NUMBERS  INTO   TWO  PARTS.       29 
THE   NUMBER   FOUR. 


Four  may  be  divided  into  :    A.    Three  and  one. 
B.  Two  and  two.     It  follows  that 

B. 


A.     a.  3  +  i  =  4- 


c.  4  —  1= 
^.4—3  = 


a.  2  +  2=4. 
£.4  —  2  =  2. 

c.  2  X  2  =  4. 

d.  \  of  4  =  2. 


e.  2  in  4  =  2  times. 


THE   NUMBER   FIVE. 


This  may  be   separated   into  :    A.   Four  and  one. 
B.  Three  and  two.     It  follows  that 


A.     0.  4  +  i  =  5. 

b.  1+4  =  5- 

c.  5-1-4. 

d.  5-4=1- 


B.     a. 


2  =  5. 
^.2  +  3=5. 

c.  5-2-3- 

d.  5-3=2. 


THE   NUMBER    SIX. 


)  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

Six  may  be  divided  as  follows  : 

A.    Five  and  one. 

a.   $  +  1=6.  c.  6  —  i  =  5. 

£.1  +  5=6.  dl  6  —  5  =  i. 

B.    Three  and  three, 
tf .  3  +  3  =  6.  ^.3x2=6. 

£.6-3  =  3.  *.  Jof6  =  3. 

^.2x3=6.  /.  j-  of  6  =  2. 

C.  Four  and  two. 

a.  4  +  2  =  6.  d.  6  —  4  =  2. 

£.2+4  =  6.  <?.  3  x  2  =  6. 

c.  6  —  2  =  4.  f.  J  of  6  =  2. 


THE   NUMBER   SEVEN. 


Seven  may  be  divided  into  : 

A.  Six  and  one. 

#.  6  +  i  =  7.  ^.7—1=6. 

b.   1+6  =  7.  d.  7  —  6=  i. 

B.  Five  and  two. 

#.   5  +  2  =  7.  ^.7  —  2  =  5. 

£.2  +  5=7.  d.  7-5=2. 

C.    Four  and  three. 

a.  4+3  =  7-  c.   7-3=4- 

^  3  +  4  =  7-  ^.7-4  =  3. 


SEPARATING  NUMBERS  INTO  TWO  PARTS.       31 
THE   NUMBER   EIGHT. 


Eight  may  be  divided  into  : 

A.  Seven  and  one. 

#.7+1=8.  c.  8  --  i  =  7. 

A   i  +7  =  8.  d.  8-7=  i. 

B.  Four  and  four. 

a.  4  +  4  =  8.  c.  2  X  4  =  8.         ^.4X2 

£.8-4  =  4.          d.  £of8=4.        /  Jof8 

C.  Five  and  three. 

tf.   5  +  3  =  8.  <:.  8  •-  3  =  5. 

b.  3  +  5  =  8.  rf.  8  -  5  =  3. 

D.  Six  and  two. 

#.  6  +  2  =  8.  £.8  —  2=6. 

£.2  +  6  =  8.  rf.  8  —  6  =  2. 


THE   NUMBER   NINE. 


32  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

Nine  may  be  divided  into  : 

A.  Eight  and  one. 

#.  8  +  i  =  9.  <:.  9  —  i  =  8. 

£.1+8=9.  d.g—8  =  i. 

B.   Five  and  four. 


a.   5+4-9. 
6.  4+5=9- 


c.  9-4-5. 
d>  9-5=4- 


C.  Six  and  three. 

#.6  +  3=9.  d.  9  —  6  =  3. 

£.3+6=9.  ^3><3=9- 

*.  9  -  3  =  6.  /  \  of  9  =  3. 

D.  Seven  and  two. 


a.  7  +  2=9. 

b.  2  +  7  =  9. 


c.  9  -  2  =  7. 

d.  9-7  =  2. 


THE   NUMBER   TEN. 


e        • 


r 


e  • 


SEPARATING  NUMBERS  INTO  TWO  PARTS.       33 


Ten  may  be  divided  into  : 

A.  Nine  and  one. 

a.  9  +  i  =  10.  c.   io  —  i  =  9. 

A   1+9=10.  rf.   io  —  9=1. 

B.  Five  and  five. 

a.  5  +  5  =  10.  c.  2  X    5  =  10. 

b.  10—5=    5.  d.  \  of  10=    5. 

C.  Six  and  four. 

a.  6  +  4  =  10.  £.10  —  4  =  6. 

£.4  +  6=10.  ^.10  —  6=4. 

D.  Eight  and  two. 

a.  8  +  2  =  10.  d.   io  —  8=    2. 

b.  2  +  8  =  io.  e.     5  X  2  =  io. 

c.  io  —  2=    8.  /  £of  10=    2. 

<m 

E.   Seven  and  three. 
a.  7  +  3  =  io.  c.   10-3  =  7. 

£.3  +  7  =  10.  ^.10—7  =  3. 


FURTHER    USE    OF    CHART    II. 

For  further  instruction  in  regard  to  the  use  that 
can  be  made  of  Chart  II.,  I  will  explain  some  addi- 
tional work  upon  the  number  eight.  Arrange  eight 
balls  on  the  numeral  frame,  as  shown  below. 


o     e 
o     • 


34  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

There  are  four  balls  at  the  left  and  four  balls  at 
the  right ;  or,  there  are  four  balls  above  and  four 
balls  below.  So,  4  and  4  are  8.  Four  balls  are  to 
be  seen  two  times  ;  therefore,  2  times  4  balls  are 
equal  to  8  balls.  Two  balls  appear  always  under  two 
other  balls  ;  hence  4X2  —  8. 

If  we  take  four  balls  from  the  eight  balls,  then 
four  balls  remain  :  hence  8—4  —  4.  This  removal 
may  be  indicated  by  covering  part  of  the  balls. 
The  line  across  the  rectangle  divides  the  dots  into 
two  equal  parts.  This  may  be  shown  on  the  frame 
by  holding  a  pointer  between  the  two  fours.  It  fol- 
lows that  the  half  of  8  is  four.  Two  balls  appear 
four  times  ;  therefore  the  fourth  part,  or  a  fourth,  of 
eight  is  two. 

These  considerations  prepare  for  the  following 
questions :  How  many  are  4  +  4?  2X4?  4X2? 
8  —  4  ?  \  of  8  ?  \  of  8  ?  What  number  must  one  put 
with  4  to  make  eight  ?  How  many  more  is  8  than 
4  ?  How  many  less  than  8  is  4  ?  How  many  times 
4  is  8  ?  How  many  times  2  is  8  ?  Of  what  number 
is  4  the  half  ?  Of  what  number  is  2  the  fourth  ? 
What  part  of  8  is  4  ?  What  part  of  8  is  2  ?  How 
many  is  8  less  2X2?  How  many  is  8  less  3X2? 
How  many  times  2  is  8  less  4  ?  etc. 

These  exercises  with  pure  numbers  are  the  proper 
preparation  for  such  simple  practical  examples  as 
these :  Charles  has  4  cents,  and  Fred  has  4  cents ; 
how  many  have  they  together  ?  Charles  has  8  cents, 


SEPARATING  NUMBERS  INTO   TWO  PARTS.       3$ 

and  gives  4  of  them  to  Fred ;  how  many  has  Charles 
left  ?  Charles  got  4  cents  yesterday,  and  4  more 
to-day ;  how  many  times  4  cents  has  he  ?  How 
many  cents  in  all  ?  Charles  had  8  cents,  and  lost 
half  of  them  ;  how  many  has  he  now  ?  Charles  and 
Fred  together  had  8  apples,  and  divided  them  so  that 
each  had  an  equal  number ;  how  many  did  each  then 
have  ?  Each  of  4  children  had  2  pears ;  how  many 
had  they  all  together  ?  Four  children  divide  8  apples 
equally  among  them  ;  how  many  does  each  receive  ? 
—  Give  a  boy  8  pencils,  and  let  him  give  one  each  to 
4  other  boys,  and  then  one  more  to  each  of  them. 
What  is  \  of  8  ?  —  One  apple  costs  2  cents  ;  how 
many  cents  do  2  apples  cost  ?  3  apples  ?  4  ?  A  yard 
of  ribbon  costs  8  cents ;  how  much  does  half  a  yard 
cost  ?  A  fourth  ?  Three-fourths  ?  etc.,  etc. 

In  giving  practicable  problems  it  is  often  necessary 
to  mention  coins,  measures,  and  weights.  These 
should  not  only  be  well  known,  but  they  should  often 
be  shown  to  the  children.  The  teacher  should  limit 
his  problems  to  those  coins,  weights,  and  measures 
that  are  accessible  to  the  children  in  their  ordinary 
intercourse.  The  copper,  nickel,  and  smaller  silver 
coins  are  all  the  coins  that  should  be  mentioned  in 
these  early  problems  ;  the  measures  should  be  limited 
to  the  inch,  foot,  yard,  pint,  and  quart  ;  and  the 
ounce  and  pound  weights  are  enough.  It  is  well  to 
have  all  the  measures  involved  in  the  problems  given, 
constantly  before  the  eyes  of  the  children,  so  that 


36  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

they  will  be  impressed  upon  the  memory.  As  the 
work  in  numbers  progresses,  these  illustrations  may 
be  enlarged.  Their  application  will  be  indicated  as 
we  progress. 

The  number  pictures  which  are  studied  with  the 
children  during  the  lesson  should  be  copied  upon  the 
slates  as  written  work.  The  teacher  can  at  first 
make  them  upon  the  board,  and  subsequently  have 
them  copied  from  Chart  II.  When  this  has  been 
repeated  sufficiently,  they  may  be  written  from 
memory. 

The  chart  will  also  serve  a  good  purpose  in  con- 
ducting reviews.  What  is  represented  in  the  chart 
may  be  expressed  in  words.  In  addition,  the  verbal 
expressions  would  run  thus  : 

One  and  one  are  two.  Four  and  one  are  five. 

Two  and  one  are  three.  Three  and  two  are  five. 

Three  and  one  are  four.  Five  and  one  are  six. 

Two  and  two  are  four.  Etc.,  etc.          • 

This  order  from  left  to  right  on  the  chart  is  to  be 
interchanged  with  the  movement  from  right  to  left, 
from  top  to  bottom,  from  bottom  to  top,  and  with 
exercises  out  of  order. 

By  regarding  each  picture  as  a  number,  and  cover- 
ing first  the  dots  at  the  right  and  then  those  at  the 
left,  numerous  exercises  in  subtraction  may  be  formed. 

By  means  of  these  exercises  all  the  facts  of  the  ad- 
dition, subtraction,  multiplication,  and  division  tables 


LEARNING    THE    USE    OF  FIGURES. 


37 


may  be  learned,  where  the  sum,  minuend,  product, 
or  dividend  does  not  exceed  ten.  Since  these 
results  are  of  the  greatest  use  in  all  arithmetical 
operations,  they  must  be  firmly  fixed  in  the  mem- 
ory. This  is  to  be  done,  however,  by  observing  and 
stating  the  facts  as  shown  on  the  chart,  by  copying 
the  number  pictures,  and  by  written  representations 
in  figures,  not  by  learning  the  statements,  as  such,  by 
heart. 

8.    LEARNING  THE  USE  OF  FIGURES. 

The  ground  already  covered  is  sufficient  for  a 
fourth  of  a  year,  and,  under  some  conditions,  for  a 
longer  time.  Hitherto  the  children  have  learned 
only  from  observation  ;  now,  however,  they  may 
without  danger  pass  from  things  to  signs,  from  num- 
bers to  figures.  This  transition  may  be  made  by 
means  of  the  following  chart : 

CHART    III. 

USE  OF  FIGURES. 


6 


8 


9 


1O 


WRITTEN  FIGURES. 


5k 


4 


•  o  • 

©  0  • 


6 


7 


9 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


Through  diligent  copying,  pointing,  and  reciting, 
the  children  will  impress  these  forms  upon  the  mind 
so  that  they  can  be  made  without  a  copy. 

The  different  rectangles  of  Chart  II.  may  then  be 
copied,  and  with  them  the  corresponding  figures  may 
be  copied  in  similar  rectangles,  as  follows  : 


•          • 

1      1 

• 

2        1 

•  •  •      • 

3        1 

* 

• 

2        2 

•     • 
• 
•     • 

4        1 

•  O  • 

•• 

3        2 

The  rest  of  the  chart  may  be  treated  in  the  same 
way. 

9.   WRITTEN  EXERCISES  IN  ADDITION. 

In  order  still  further  to  represent  in  figures  the 
results  gained  through  the  preceding  instruction,  we 
make  use  again  of  Chart  II.,  at  first  for  the  purpose 
of  addition.  The  children  must  now  learn  the  sign 
of  addition  (+,  plus,  or  and)  and  also  the  sign  of 
equality  (=,  is,  or  are).  Then,  by  writing  the  figures 
for  the  dots  seen  in  the  different  rectangles,  they  can 
form  these  series  of  numbers  on  their  slates  : 


1  +  i  =  2,  or  i  +  i  =  2. 

2  +  i  =  3,  or  i  +  2  =  3. 

3  +  i  =  4,  or  i  +  3  =  4. 


3  +  2  =  5,  or  2  +  3  =  5. 
5  +  i  =  6,  or  i  +  5  =  6. 
3  +  3  =  6,  or  3  +  3  =  6. 


WRITTEN  EXERCISES  IN  ADDITION.  39 

2  +  2=  4,  or  2  +  2=  4.     4  +  2  ^=  6,  or  2  +  4  =  6. 
4  +  i  =  5,  or  i  +  4  —  5.  etc.     etc. 

At  first  the  work  may  be  confined  to  a  few  of  the 
number  pictures,  but  it  should  be  extended  gradu- 
ally till  the  whole  chart  can  be  represented  in  fig- 
ures. 

In  order  to  teach  regularity  and  order  in  the 
arrangement  of  the  figures,  it  is  worth  while  to  have 
the  slates  ruled  upon  one  side  in  squares  of  about 
three-eighths  of  an  inch  ;  the  other  side  may  be  ruled 
in  lines  for  writing.  After  the  first  year  the  squares 
may  be  omitted,  but  at  first  they  are  very  helpful. 
A  portion  of  the  blackboard  should  be  ruled  in  the 
same  way. 

When  Chart  II.  can  be  readily  interpreted  in  this 
way  by  figures,  the  reverse  process  should  be  intro- 
duced. The  children  should  be  required  to  trans- 
late figures  into  numbers.  The  teacher  will  write 
upon  the  board,  for  example,  3+2  =  5,  and  the 
children  will  copy  the  same,  and  then  add  the  cor- 
responding number  pictures,  thus  : 


For  a  review,  and  for  drill  in  this  work,  the  upper 
part  of  Chart  IV.  furnishes  a  convenient  means. 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


CHART    IV. 

WRITTEN  REPRESENTATION  OF  CHART  II. 


1  +  1=2 

3  +  2 

5  +  2 

6  +  2 

9  +  i 

2+1  = 

5  +  i 

4  +  3 

8+1 

5  +  5 

3  +  i- 

3  +  3 

7  +  i 

5+4 

6  +  4 

2  +  2  = 

4  +  2 

4  +  4 

6  +  3 

8+2 

4+1  = 

6  +  1 

5  +  3 

7  +  2 

7  +  3 

2  —  1  =  1 

S-2 

7-2 

8-2 

10  —  I 

3->  = 

6-1 

7-3 

9-1 

10-5 

4-i  = 

6-3 

8-1 

9-4 

10  —  4 

4-2  = 

6-2 

8-4 

9-3 

10  —  2 

5'-I  = 

7-i 

8-3 

9-2 

10-3 

1C.   WRITTEN  EXERCISES  IN  SUBTRACTION. 

For  the  first  practice  in  written  subtraction,  Chart 
II.  may  be  used.  The  children  must  learn  the  mean- 
ing of  the  sign  of  subtraction  (— ,  less)  and  use  this 
in  representing  the  results  of  their  observation.  By 
observing  all  the  dots  in  the  rectangles,  and  then 
covering  first  those  in  the  right  and  then  those  in 
the  left,  the  following  results  will  be  reached : 

A.  2— 1  =  1,     4  —  2  =  2,     B.  2  —  1  =  1,     4  —  2  =  2, 
3~i  =  2,     5-1=4,  3-2-1,     5-4=1, 

4-i  =  3»     5-2  =  3,          4-3  =  i»     5-3  =  2, 
etc.  etc.  etc.  etc. 


WRITTEN  EXERCISES  IN  SUBTRACTION. 


When  Chart  II.  can  be  observed,  and  the  corre- 
sponding figures  readily  written,  the  process  should 
be  reversed.  The  children  should  produce  the  num- 
bers when  the  figures  are  shown.  The  work  on  the 
pupils'  slates  may  assume  this  form. 

5-2-3- 

The  lower  part  of  Chart  IV.  furnishes  a  convenient 
means  of  drill  in  the  interpretation  of  figures  denot- 
ing subtraction. 

As  a  final  review  of  this  kind  of  work,  Chart  V. 
will  be  useful,  inasmuch  as  it  requires  the  pupil  to 
interpret  the  signs  +  and  — ,  as  well  as  to  indicate 

operations. 

CHART    V. 

FOR  REVIEW  OF  CHART  II. 


7+i 

8-5 

3-i 

2  —  I 

5-3 

7-2 

IO—9 

2  +  1 

2  +  6 

5-1 

9-8 

6+1 

4-1 

10-3 

4  +  4 

8-2 

6-5 

4  +  5 

2  +  8 

5  +  3 

10-7 

8  +  1 

9-4 

8-1 

6  +  3 

i  +  i 

8+2 

3  +  7 

8-4 

2  +  2 

i  +  7 

3-2 

6-1 

4  +  6 

10  —  8 

i  +  5 

2+5 

6-4 

9-5 

7-  i 

5-2 

10  —  2 

8-3 

3  +  i 

i+4 

6-2 

1+6 

3+3 

io  —  6 

5  +  2 

9-7 

5+4 

9+i 

2+4 

2  +  7 

2+3 

5-4 

1  +  2 

5  +  i 

5  +  5 

3+2 

7-5 

10—4 

4+i 

6+2 

3+4 

3  +  5 

7  +  3 

4  +  2 

6+4 

8-7 

8-6 

7-4 

i+3 

3  +  6 

i+9 

10-5 

1  +  8 

4  +  3 

7-3 

7-6 

7  +  2 

6-3 

9-6 

42  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

9 

11.  COMBINED  ADDITION  AND  SUBTRACTION. 

Up  to  this  point  the  written  exercises  have  been 
connected  immediately  with  the  observation  of  the 
chart.  Nothing  more  has  been  required  of  the  chil- 
dren than  the  translating  of  the  number  pictures  into 
figures,  and  figures  into  number  pictures.  Now,  in 
order  to  free  the  written  work  from  the  necessity 
of  observation  ;  to  replace  immediate  knowledge  of 
objects  with  ideas  of  objects,  the  results  of  the  addi- 
tions and  subtractions  may  be  united  in  the  same 
written  exercises,  so  that  the  one  may  furnish  the 
clew  to  the  other,  thus  : 

2+1=3.  3-1—2. 

1+2  =  3.  3  —  2  =  1. 

The  whole  of  Chart  V.  may  be  treated  in  this  way. 

While  exercises  in  multiplication  and  division  have 
not  been  hitherto  excluded,  they  are  not  numerous 
enough  in  this  stage  to  make  it  worth  while  to  intro- 
duce them  into  the  written  work  as  special  topics. 

Before  proceeding  to  explain  the  treatment  of  num- 
bers in  the  following  stage  of  the  work,  I  will  remark 
that  it  is  of  the  utmost  importance  that  the  work  in 
numbers  from  one  to  ten  should  be  thoroughly  mas- 
tered. Naming  any  number  up  to,  and  including, 
ten,  and  also  one  part  of  the  number,  should  instantly 
suggest  to  the  child  the  other  part.  The  two  parts 


COMBINED  ADDITION  AND   SUBTRACTION.      43 

of  each  number  should  be  so  associated  with  each 
other  and  with  the  number  that  one  part  cannot  be 
thought  of  as  such  without  the  idea  of  the  other  part 
being  at  once  called  to  mind.  Haste  here  is  not  to 
be  desired.  The  results  must  be  lastingly  fixed,  and 
this  can  only  be  accomplished  by  much  patient,  atten- 
tive, earnest  effort. 

I  have  suggested  a  progressive  use  of  a  few  kinds 
of  apparatus,  but  I  would  by  no  means  limit  the 
teacher  to  these.  Variety  of  illustration  is  desirable  ; 
but  it  is  also  desirable  to  have  some  means  of  making 
the  children  do  such  work  as  will  cause  the  desired 
results,  which  will  not  be  a  constant  drain  upon  the 
teacher's  power.  Hence  the  free  use  of  the  charts 
for  review  is  recommended. 


44  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

CHAPTER   II. 

NUMBERS  FROM  ONE  TO  TWENTY. 
12.    COUNTING  TO  TWENTY. 

As  in  teaching  numbers  from  one  to  ten  we  began 
with  counting,  so  we  do  in  teaching  numbers  from 
ten  to  twenty.  Put  ten  balls  on  the  upper  wire  of 
the  numeral  frame.  Let  the  children  find  how  many 
twos  there  are  in  ten,  how  many  fives,  how  many  tens, 
and  how  many  ones.  Then  tell. them  that  ten  ones 
are  called  a  ten,  and  that  one  is  called  a  unit.  Count 
out  ten  units  on  the  upper  wire,  and  call  the  result 
one  ten.  Put  one  ball  out  on  the  second  wire ;  then, 
pointing  first  to  the  ten  and  next  to  the  one,  say : 
"One  ten  and  one  unit  make  eleven  units."  Add 
another  ball,  and  then,  pointing  as  before,  say  :  "  One 
ten  and  two  units  are  twelve  units."  And  so  pro- 
ceed to  the  sentence  :  "  One  ten  and  nine  units  make 
nineteen  units." 

Add  another  ball,  and  there  appear  two  rows  of 
ten  each,  thus  : 


•     o     •     o 


The  truth  which  the  pupils  gain   from  observing 


COUNTING    TO    TWENTY.  45 

these  balls  is  expressed  :  Two  rows  are  two  tens,  or 
twenty  units.  These  dots  should  then  be  copied  by 
the  children  on  their  slates  from  a  copy  made  on  the 
board  by  the  teacher. 

For  further  practice  let  the  above  sentences  be 
repeated  as  the  balls  are  shown,  from  ten  to  twenty ; 
and  then  let  the  counting  from  one  to  twenty  be 
practised,  introducing  the  following  changes  : 

1.  One,  two,  three,  four,  five,  and  so  on  to  twenty. 

2.  After  one  comes  two,  after  two  comes  three, 
and  so  on  to  twenty. 

3.  One  and  one  are  two,  two  and  one  are  three, 
and  so  on. 

4.  One,  three,  five,  seven,  etc. 

5.  Two,  four,  six,  eight,  etc. 

6.  One,  four,  seven,  ten,  etc. 

7.  Two,  five,  eight,  eleven,  etc. 

8.  Three,  six,  nine,  twelve,  etc. 

9.  Twenty,  nineteen,  eighteen,  etc. 

10.  Before  twenty  comes  nineteen,  etc. 

11.  Twenty  lees  one  is  nineteen,  etc. 

12.  Twenty,  eighteen,  sixteen,  etc. 

13.  Nineteen,  seventeen,  fifteen,  etc. 

14.  Twenty,  seventeen,  fourteen,  etc. 

When  the  children  can  surely  and  readily  perform 
these  exercises  ;  can  unite  a  ten  and  a  fundamental 
number  —  that  is,  a  number  from  one  to  ten;  can 
change  any  number  from  eleven  to  twenty  into  tens 


46  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

and  units  ;  and  when  they  can,  further,  construct  any 
number  on  their  slates  by  arranging  the  proper  dots 
in  tens  and  units  ;  and  can  name  any  number  shown 
them  by  balls  or  marks, — then  and  not  till  then,  may 
they  be  allowed  to  pass  on  to  the  representation  in  fig- 
ures of  numbers  from  eleven  to  twenty.  Till  they 
have  reached  this  ability,  they  may  be  kept  practising 
upon  the  written  work  connected  with  numbers  from 
one  to  ten.  This  will  constitute  a  valuable  review. 

Written  work  should  never  precede  corresponding 
oral  work ;  for  the  written  work,  at  this  stage,  is 
simply  designed  to  impress  upon  the  mind  what  the 
oral  work  has  already  made  clear  to  the  understand- 
ing. It  is  useful  for  review,  but  should  keep  a  few 
steps  behind  the  oral  work,  whenever  the  children 
.are  introduced  to  a  new  topic.  Written  work  de- 
mands more  self-independence ;  but  in  classes  com- 
posed of  several  divisions  the  pupils  must  be  thrown 
more  upon  their  own  resources.  On  account  of 
the  weaker  children,  therefore,  the  written  exercises 
should  be  deferred  till  a  perfect  understanding  is 
gained  and  a  certain  degree  of  facility  is  reached.  It 
is  well  to  bear  this  remark  in  mind  constantly. 

The  written  representation  of  numbers  from  eleven 
to  twenty  is  not  difficult  for  children  to  comprehend. 
The  figure  standing  for  the  ten  is  put  at  the  left,  that 
representing  the  units  at  the  right,  therefore : 
I  ten  and  I  unit    =  1 1  units. 
I  ten  and  2  units  =  12  units. 


SEPARATING  NUMBERS  INTO   TWO  PARTS.      47 

I  ten  and  3  units  =  13  units,  and  so  on  to  nineteen. 
The  following  series  may  be  explained  and  copied  : 

10+1  =  11.         10+    8  =  18.          14+1  =  15. 

10  +  2  =  12.  10+     9=19.  IS  +  I  =  I6. 

10+3  =  13.  10+10  =  20.  16+1=17. 

10  +  4=14.         10+    1  =  11.         17+1  =  18. 

10+5  =  15.  11+      1=12.  18+1  =  19. 

10  +  6=16.         12+    1  =  13.         19+1=20. 
10  +  7  =  17.         J3  +    i  =  T4- 


13.  SEPARATING  NUMBERS  INTO  Two  PARTS. 

In  general  the  same  course  is  to  be  followed  in  the 
division,   or  separation,  of    numbers  from   eleven  to 


CHART    VI. 

UNITING  AND  SEPARATING  NUMBERS  FROM  ELEVEN  TO  TWENTY. 


«o  •• 

•Q  •• 


e©  •• 


999 


••  Q9 


9  •  • 


0  G 


9  9 

9 
®  Q 


9®  99 

99  99 


•e  •• 


•0  09 

•  c 


9  9 
9  9 


9  9  © 


•  O  99 
99  99 


99  CO 
•  © 
•  O  ©© 


0   • 


•   O 
99  •« 


99  99 
9   9 

eo  •• 


48  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

twenty  as  was  recommended  in  regard  to  numbers 
from  one  to  ten.  The  results  needed  in  all  the  fun- 
damental rules  may  be  obtained  by  observing  the 
separation  of  the  several  numbers  into  the  various 
pairs  of  which  they  are  composed,  as  shown  on  the 
preceding  chart. 

The  treatment  of  numbers  at  this  stage  of  the 
work  is  almost  the  same,  in  general,  as  in  the  case  of 
numbers  below  ten.  The  division  of  each  number 
between  eleven  and  twenty  is  to  be  indicated  by  the 
arrangement  of  the  balls  on  the  numeral  frame  by 
the  teacher.  From  each  division  two  results  in  addi- 
tion and  two  in  subtraction  are  to  be  obtained  ;  and, 
in  the  case  of  numbers  composed  of  factors,  at  least 
one  result  for  division  and  one  for  multiplication. 
No  division  is  to  be  made,  however,  that  will  make 
one  part  of  the  divided  number  greater  than  ten. 
The  division  of  the  numbers  here  meant  is  simply 
the  separation  of  the  numbers  into  two  parts. 

The  divisions  are  to  be  carefully  shown,  one  after 
another,  the  number-pictures  made  by  the  children, 
and  the  truth  stated  orally,  with  frequent  repetition, 
before  the  chart  is  called  into  use.  The  chart  is  for 
review  only  :  first,  by  reciting  the  facts  as  shown  by 
the  arrangement  of  dots ;  second,  by  copying  the 
number-pictures  on  the  slates  ;  and,  third,  by  repre- 
senting the  dots  by  figures. 

I  will  first  indicate  the  results  to  be  reached,  and 
then  make  suggestions  as  to  the  manner  of  doing  the 
work. 


SEPARATING  NUMBERS  INTO  TWO  PARTS.      49 
THE   NUMBER   ELEVEN. 

Eleven  can  be  divided  into : 

a.  Ten  and  one  ;  hence, 

10+    i  =  n,  ii  —    i  —  IO, 

1  +  io— ii,  ii  — 10=    I. 

b.  Six  and  five  ;  hence, 

6+5  —  11,  11  —  5=6, 

5+6  —  n,  11—6  —  5. 

c.  Nine  and  two  ;  hence, 

9  +  2  — ii,  11  —  2—9, 

2  +  9—11,  11—9  —  2. 

d.  Eight  and  three  ;  hence, 

8  +  3-1 1,  11-3-8, 

3  +  8-n,  11-8-3. 

e.  Seven  and  four ;  hence, 

•     7  +  4-1 1,  11-4-7, 

4  +  7=n>  11-7=4. 

THE   NUMBER  TWELVE. 

Twelve  can  be  divided  into  : 

a.  Ten  and  two  ; 

10+     2  —  12,  12—     2  —  IO, 

2  +  IO—  12,  12—  IO—     2. 

b.  Six  and  six  ; 

6  +  6—12,  6  in  12  two  times, 

12—6—6,  6  X       2  —  12, 

2  X  6  —  12,  \  Of  12  —     2, 

\  of  1 2  —    6,  2  in  1 2  six  times. 


5O  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

c.  Eight  and  four ; 

8  +  4=  12,  12  —    8  =    4, 
4+8-12,  3  X    4-12, 

12-4=    8,  \  of  12=   4, 

4  in  12  three  times. 

d.  Nine  and  three  ; 

9  +  3  =  12,  12-9-3, 
3  +  9-12,  4x3-12, 

12-3-9,  i of  12-    3, 

3  in  12  four  times. 

*.  Seven  and  five  ; 

7  +  5-12,  12- 5 -7, 
5 +  7- 12,  12-7-5. 

THE   NUMBER   THIRTEEN. 

Thirteen  can  be  divided  into : 

a.  Ten  and  three  ; 

10+    3  -  13,  13—    3  =  10, 

3  +  10-13,  13-10-    3. 

b.  Eight  and  five  ; 

8  +  5    =13,  i3-5=8> 

5  +  8    -13,  13-8-5. 

c.  Nine  and  four ; 

9  +  4    =•  13,  13-4-9, 

4  +  9    =  I3>  13-9=4- 

d.  Seven  and  six  ; 

7  +  6    =13,  13-6  =  7, 

6  +  7    =13,  13  —  7  =-6. 


SEPARATING  NUMBERS  INTO  TWO  PARTS.       5 1 
THE  NUMBER  FOURTEEN. 

Fourteen  can  be  divided  into : 

a.  Ten  and  four ; 

10  +  4,       4+io,     14  —  4,     14—10. 

b.  Seven  and  seven  ; 

7  +  7,     14  —  7,         2x7,      7  in  14,  i  of  14. 

c.  Eight  and  six ; 

8  +  6,  14-8,  \  of  14, 
6  +  8,                   7X2,  2  in  14. 

14  —  6, 

d.  Nine  and  five  ; 

9+S>     5  +  9>     H- 5>     I4-9- 

THE  NUMBER  FIFTEEN. 

Fifteen  can  be  divided  into : 

a.  Ten  and  five  ; 

10+    5,  15-  10,  iof  15, 

5  +  10,  3  X    5>  5  in  15. 

IS-    5> 

b.  Nine  and  six ; 

9  +  6,  15-9,  iofis, 
6 +  9.                  5  x  3>  3  in  15. 

15-6, 

c.  Eight  and  seven  ; 

8  +  7,     7  +  8,     15-7.     IS-8. 


52  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

THE    NUMBER    SIXTEEN. 

Sixteen  can  be  divided  into : 

a.  Ten  and  six  ; 

10  +  6,         6+10,         16  — 6,         16—  10. 

b.  Eight  and  eight ; 

8  +  8,         8X2,         4X4,          4  in  16. 
16  — 8,         \  of  16,         2  in  16, 

2X8,         |  of  16,        i  of  16, 
£.  Nine  and  seven ; 

9  +  7,         7  +  9>  16-7,         16-9. 

THE   NUMBER   SEVENTEEN. 

Seventeen  can  be  divided  into  : 

a.  Ten  and  seven  ; 

10  +  7,         7+io,         I7~7>         17-10. 

b.  Nine  and  eight ; 

9  +  8,         8+9,         17-8,         17-    9. 

THE  NUMBER  EIGHTEEN. 

Eighteen  can  be  divided  into  : 

a.  Ten  and  eight ; 

10+    8,         18— 10,         9  X    2,          9  in  18, 
8+10,         18—    8,         |ofi8,         2  in  18. 

b.  Nine  and  nine  ; 

9  +  9,  9  in  1 8,  6  X  3, 

18-9,  3X6,  iofiS, 

2x9,  \  of  18,  6  in  18. 

\  of  18,  6  in  18, 


NUMBERS  FROM  ELEVEN  TO    TWENTY.         $3 
THE   NUMBER  NINETEEN. 

Nineteen  can  be  divided  into : 
Ten  and  nine ; 

10  +  9,         9+io,         19  —  9,         19  —  10. 

THE  NUMBER  TWENTY. 

Twenty  can  be  divided  into : 

Ten  and  ten ; 

10  +  10,  10  in  20,  5  X  4, 

20  —  10,  4XS,  i  of  20, 

2  X  10,  \  of  20,  4  in  20. 

\  of  20,  5  in  20, 

14.  TEACHING  NUMBERS  FROM  ELEVEN  TO 
TWENTY. 

It  has  already  been  remarked  that  it  is  of  the 
highest  importance  for  the  pupils  to  know  every  two 
parts  of  which  each  number  from  one  to  ten  consists. 
As  an  indication  of  the  way  the  work  in  developing  a 
knowledge  of  numbers  from  eleven  to  twenty  should 
be  managed,  I  will  show  by  a  few  examples  how  to 
utilize  this  knowledge  of  the  parts  of  the  fundamental 
numbers. 

If  5  is  to  be  added  to  8,  let  2  be  added  first,  so  as 
to  make  10.  If  this  2  be  taken  from  the  5,  3  remains ; 
and  this  3  added  to  10  makes  thirteen ;  therefore,  5 
added  to  8  makes  13.  In  general,  first  add  enough 


54  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

to  make  10 ;  then  add  the  rest  of  the  number  to  be 
added. 

If  5  is  to  be  subtracted  from  13,  first  subtract  3,  so 
that  the  remainder  will  be  10;  then  from  the  10  take 
away  the  rest  of  the  5,  namely,  2,  and  the  remainder 
is  8. 

The  relations,  or  truths,  shown  by  the  number 
pictures  for  8  and  5,  may  be  indicated  as  follows : 

8  +  5  may  be  resolved  into    8  +  2  —  10; 
10  +  3-13. 

5  +  8  may  be  resolved  into    5  +  5  =  10; 
10  +  3-13. 
13  —  5  may  be  resolved  into  13  —  3  —  10; 

10  —  2—     8. 

13  —  8  may  be  resolved  into  13  —  3  —  10. 
10-5-    5- 
The  numbers  9  and  7  may  be  treated  thus  : 

9  +  7  may  be  changed  into    9  +  i  —  10 ; 
10  +  6—  16. 

7  +  9  may  be  changed  into    7  +  3  —  10 ; 
10  +  6  —  16. 

16  —  7  may  be  changed  into  16  —  6  —  10  ; 
10—  i  —    9. 

16  —  9  may  be  changed  into  16  —  6  —  10 ; 
10-3=    7- 

These  processes  and  results  are  first  to  be  shown 
by  means  of  the  balls  on  the  numeral  frame,  then  by 
the  number  pictures,  which  are  first  to  be  made  by 


NUMBERS  FROM  ELEVEN  TO    TWENTY.         55 

the  teacher  on  the  board  and  afterwards  copied  by 
the  children  on  the  slates.  As  fast  as  the  number 
pictures  have  been  treated  in  this  way,  Chart  VI.  may 
be  used  as  a  means  of  review. 

The  chart  is  to  be  read  from  left  to  right,  right  to 
left,  top  to  bottom,  and  bottom  to  top.  If  the  child 
hesitates  in  this  reading,  the  teacher  should  lead  him 
to  see  the  divisions  of  the  numbers  to  be  added  or 
subtracted,  such  that  the  results  first  obtained  will 
always  be  10.  By  this  means  the  pupil  will  learn  to 
think  to  the  desired  result  without  counting.  When 
this  reading  of  the  chart  can  be  gone  through  with 
rapidly  and  correctly,  the  chart  may  be  copied  picture 
by  picture,  thus  : 


10+    i  =  u,         11  —  10=    i, 

I  +  IO  =  II,  II  -       I  =  IO. 

Next  should  follow  the  reverse  of  this  process, 
namely,  writing  the  corresponding  number  pictures 
when  the  figures  are  given.  On  Chart  VII.  are  the 
figures  corresponding  to  the  number  pictures  on  Chart 
VI.  Let  the  pupils  copy  these  figures,  and  at  first 
produce  the  corresponding  pictures  ;  but  later  the 
results  may  be  written  immediately  in  figures,  or 
recited  orally. 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


CHART    VII. 

WRITTEN  REPRESENTATION  OF  CHART  VI. 


10+  I 

6+6 

9  +  4 

io+5 

10+7 

6+5 

8+4 

7+6 

9  +  6 

9  +  8 

9+  2 

9+3 

10+4 

8+7 

10  +  8 

8+3 

7+5 

7+7 

10+6 

9  +  9 

7  +  4 

io+3 

8+6 

8+  8 

10  +  9 

IO  +  2 

*+5 

9+5 

9+7 

IO  +  IO 

II  —  I 

!2-  6 

i3-4 

I5~  5 

17-7 

"-5 

12-4 

13-6 

15-6 

17-8 

II  —  2 

12-3 

14-4 

I5~  7 

18-8 

"-3 

12-5 

14-  7 

16  —  6 

18  —  9 

ii  —  4 

13-3 

14  -  6 

16-  8 

19-9 

12—2 

!3~5 

M-5 

16-7 

20   -  10 

Charts  VIII.  and  IX.  are  designed  to  assist  in  the 
final  review  of  the  addition  and  subtraction  of  num- 
bers from  one  to  twenty.  This  work  completes  the 
learning  of  the  tables  of  addition  and  subtraction, 
which  was  begun  in  sections  9  and  10,  and  hence 
should  be  made  very  thorough.  These  charts  should 
be  copied  by  the  children.  Sometimes  the  corre- 
sponding pictures  should  be  constructed,  and  some- 
times the  results  should  be  written  at  once  in  figures. 
The  drill  should  be  partly  oral ;  at  one  time  the  pupil 
reading  from  the  chart  and  giving  the  result ;  at  an- 
other, the  teacher  should  read.  The  work  may  be 


TEACHING  NUMBERS. 


57 


varied  by  letting  the  reading  and  reciting  both  be 
done  by  pupils. 


CHART    VIII. 

FOR  REVIEW  OF  CHART  VI. 


16-  6 

9+5 

12-7 

9+6 

3  +  10 

10  +    I 

i5-5 

10+  3 

9+8 

2  +  IO 

8+4 

ii  -4 

6+5 

18-  8 

17  —  10 

13-3 

12-6 

12-4 

19-9 

14  —  10 

I2-5 

14  —  8 

II  —  I 

7+5 

16  —  10 

9+  2 

8-8 

4+7 

12  —  9 

7  +  10 

13-8 

10+6 

9  +  4 

IO  +  2 

I  +  10 

5+7 

17-  7 

13-9 

12-8 

8  +  10 

i3-5 

9+3 

II  —  2 

6+6 

19  —  10 

3+8 

ii-  3 

8  +  6 

8+5 

6  +  10 

CHART    IX. 

FOR  REVIEW  OF  CHART  VI. 


4+  9 

12-3 

11-9 

14-4 

18  —  10 

ii  ~5 

7  +  4 

14—6 

7+7 

II  —  IO 

12—2 

4+8 

10+  9 

2  +  9 

4+10 

5  +  9 

5  +  8 

i5-9 

14-9 

15  —  10 

15-6 

13-  7 

16-  8 

6+  8 

10+  10 

n-8 

5  +  6 

3+9 

6+9 

5  +  1° 

ii-  7 

10+4 

ii-  6 

10+7 

20  —  10 

io+5 

6+7 

13-6 

17-9 

12    -  10 

18-9 

14-  7 

8+3 

10+8 

9  +  10 

9+  9 

i4-5 

7+6 

17-8 

13  -  I0 

58  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

If  you  exercise  the  children  upon  numbers  from  ten 
to  twenty,  as  was  recommended  in  regard  to  numbers 
from  one  to  ten,  namely,  by  having  the  results  of  the 
additions  and  subtractions  reached  in  all  cases  through 
the  performing  of  the  necessary  operations  upon  the 
objects  themselves,  and  then  by  having  the  results 
fixed  in  the  memory  through  the  repetition  of  the 
processes  by  which  they  were  reached,  and  not  by 
the  saying  of  the  sentences  which  express  the  results, 
you  will  have  laid  a  most  thorough  foundation  for  the 
following  stage ;  that  is,  the  treatment  of  numbers 
from  twenty  to  one  hundred.  But  not  every  child 
possesses  a  sufficiently  strong  memory  for  numbers. 
It  would  be  tiresome  to  dwell  on  this  stage  of  the 
work  till  every  child  was  perfect  in  all  the  operations 
practised.  This  perfection  is  to  be  reached  in  the 
next  stage  of  the  work,  where  the  exercises  are  simi- 
lar, where  they  are  more  varied,  and  where,  on  account 
of  their  greater  variety,  they  are  less  fatiguing. 

It  always  makes  a  difference  whether  a  child  is 
taught  alone,  or  with  many  others,  as  in  school.  In 
the  one  case,  the  work  may  be  graduated  to  the 
individual ;  but  in  school,  if  one  attempts  to  make 
the  weakest  perfect,  the  brightest,  and  even  those  of 
moderate  talent,  are  kept  back  too  much.  In  school, 
neither  the  brightest  nor  the  dullest,  but  the  average, 
must  determine  the  progress  of  the  class.  All  must 
always  be  made  to  comprehend  the  work  in  hand,  at 
least  so  far  as  is  necessary  for  understanding  what  is 


TEACHING  NUMBERS.  59 

to  follow ;  but  readiness  in  doing  may  often  be 
secured  through  the  reviews  necessarily  practised  in 
what  follows.  In  this  case  addition  and  subtraction 
of  numbers  above  twenty  will  make  imperfections 
here  disappear,  if  the  same  processes  are  continued. 

Special  attention  ought  to  be  given  to  the  numbers 
12,  15,  1 6,  1 8,  and  20,  because  they  afford  an  oppor- 
tunity to  prepare  the  children  for  multiplication  and 
division.  The  following  suggestions  are  offered  : 

\  year  =  6  months  ;      f  year  —  12  months. 

1        ft         —    A  (t  •'         %       (f         —       Q  (f 

"3  ~~  4  >         ¥ 

i  "    =3      "     ;    I  "    =  6     •• 

*    "     =2        "       ;      |    "      -    4       " 

TV    "        =   I  "         J        T22     "        =    ^          « 

If  I  apple  costs    2  cents,  what  cost  2,  3,  4,  5,  6  apples  ? 

1  3  "  2,  3, 4 apples? 
I"  4  "  2,  3  apples  ? 
6«  12  "  i,  2,  3, 4, 5  apples? 
4         "  12  "         "  i,  2,  3  apples  ? 

3         "  12          "         "          i,  2  apples  ? 

2  "  12          "         "          i  apple? 

The  written  work  on  numbers  from  ten  to  twenty 
is  to  follow  the  illustrations  of  Chart  VI. ,  as  previ- 
ously indicated. 

It  is  recommended  that  the  addition  and  subtrac- 
tion should  be  limited  to  the  fundamental  numbers, 
because  additions  in  the  second  ten  are  grounded 
upon  those  in  the  first  ten. 


60  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

If  the  child  knows  that  : 

1  +  3  =  4,  he  knows  that  11+3  =  14. 

2  +  6-8,         "  "     12  +  6—18. 

4  +  3-7,         "  "     14  +  3-17. 

It  is  only  necessary  to  call  his  attention  to  these 
facts. 


COUNTING.  6l 


CHAPTER   III. 

NUMBERS   FROM   ONE   TO   ONE   HUNDRED. 
15.   COUNTING. 

FOR  the  purpose  of  extending  the  ideas  of  numbers 
to  one  hundred  use  should  be  made  of  the  large 
numeral  frame  with  100  balls.  First  move  out  two 
rows  of  balls  on  the  frame.  These,  as  the  children 
already  know,  contain  2  tens,  or  20  units. 

Add  to  these  another  row,  and  we  have  now 

3  tens,  or  30  units  ; 

so  may  be  shown     4  tens,  or  40  units, 

5  tens,  or  50  units, 

6  tens,  or  60  units, 

7  tens,  or  70  units, 

8  tens,  or  80  units, 

9  tens,  or  90  units, 
10  tens,  or  100  units. 

The  statements  of  the  truths  thus  exhibited  may 
be  practised  by 

a.  Naming  the   numbers    in    order,   forwards   and 
backwards ; 

b.  Questioning  on  the  numbers  out  of  their  order  ; 

c.  Pointing  and  having  the  children  name ; 


62  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

d.  Naming  and  having  the  children  point ; 

e.  Forming  the  series  in  order  and  having  them 

named : 

10  +  10  =  20 ;  20  +  10  =  30,  etc. 

100  —  10  =  90 ;  90  —  10  =  80,  etc. 

The  expression  of  the  numbers  in  figures  should 
be  omitted  at  first,  so  that  the  ideas  of  the  numbers 
may  not  be  confused  with  the  figures.  If  written 
work  for  the  pupils  is  desired,  enough  may  be  found 
in  a  review  of  the  work  on  numbers  from  ten  to 
twenty. 

Now  the  teacher  may  go  back  again  to  one  ten, 
and  have  the  numbers  n,  12,  13,  and  so  on  to  20, 
formed  by  the  addition  of  one  unit  at  a  time,  as  was 
recommended  in  the  development  of  numbers  from 
ten  to  twenty.  In  the  same  way  should  the  numbers 
from  21  to  30,  31  to  40,  41  to  50,  etc.,  to  100  be 
formed.  The  numerical  facts  thus  illustrated  may 
be  expressed  : 

Twenty  and  one  are  twenty-one ; 

Twenty  and  two  are  twenty-two ; 

Twenty  and  three  are  twenty-three ; 
and  so  on  to  100;  and  also, 

Two  tens  and  one  unit  are  twenty-one  units ; 

Two  tens  and  two  units  are  twenty-two  units ; 

Two  tens  and  three  units  are  twenty-three  units, 
and  so  on  to  100. 

At  each  new  ten  the  teacher  should  stop  and  prac- 


COUNTING.  63 

tise  the  children  in  the  numbers  already  learned,  by 
questioning  them  on  the  numbers  out  of  their  order. 
For  example,  point  to  24,  27,  29,  22,  or  30  balls,  and 
ask,  How  many  balls  ?  Tell  the  pupil  to  point  to 
different  numbers.  Pointing  to  25,  ask,  How  many 
tens  and  how  many  units  are  there?  What  is  the 
number  called?  How  many  is  it  more  than  20?  How 
many  less  than  30  ?  Ask  : 

What  number  comes  after  25,  22,  29  ? 
What  number  comes  before  25,  22,  29? 
Count  forward  from  I  to  30. 
Count  backward  from  30  to  I. 
Count  2,  4,  6,  8,  10,  and  so  on  to  30. 
Count  i,  3,  5,  7,  9,  and  so  on  to  29. 
Count  30,  28,  26,  24,  and  so  on  to  o. 
Count  29,  27,  25,  23,  and  so  on  to  I. 
With  the  introduction  of  each  ten  review  from  the 
beginning. 

When  the  school  is  not  furnished  with  a  large 
numeral  frame,  a  chart  like  the  following,  Chart  X., 
may  be  used  as  a  means  of  giving  the  children  an 
intuitive  knowledge  of  numbers  from  one  to  one 
hundred,  and  of  the  decimal  system  of  numbers. 
Both  the  top  and  bottom  parts  of  the  chart  may  be 
used  for  counting  by  tens.  If  a  piece  of  stiff  paste- 
board or  a  ruler  be  cut  in  this  form,  it  may  be  so 


64  ARITHMETIC  IN  PRIMARY  SCHOOLS, 

held  as  to  cover  any  number  of  units  in  any  row ; 
and  so  by  moving  it  down  the  lower  part  of  the  chart, 
and  then  across  the  chart,  the  formation  of  all  num- 
bers from  one  to  one  hundred  may  be  shown  to  the 
eye,  the  same  as  by  the  numeral  frame. 

CHART    X. 

COUNTING  BY  TENS. 


COUNTING  TO  ONE  HUNDRED. 


Chart  X.  may  be  used  profitably  as  a  means  of 
reviewing  numbers  from  one  to  one  hundred,  either 
by  counting  by  tens  or  counting  by  units  ;  or  for 


READING  AND    WRITING  NUMBERS. 


showing  numbers  to  be  named,  or  to  be  reduced  to 
tens  and  units ;  or  for  showing  any  number  of  tens 
and  units  that  may  be  named ;  as  well  as  for  various 
other  exercises. 

16.   READING  AND  WRITING  NUMBERS. 

Written  work,  that  is,  use  of  figures,  should  not  be 
introduced  till  the  pupils  are  able  to  find  any  number 
on  the  numeral  frame  or  on  the  chart,  to  resolve  any 
number  which  may  be  shown  on  the  chart  or  frame 
into  tens  and  units,  or  to  unite  any  number  of  tens 
and  units  into  the  number  which  they  constitute.  It 

CHART    XI. 

WRITTEN  REPRESENTATION  OF  CHART  X. 


1 

2 

3 

4 

5 

.6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23  " 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

66  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

is  of  the  utmost  importance  at  this  stage  of  the  work 
that  figures  are  not  mistaken  for  numbers ;  and  to 
secure  this,  much  work  should  be  done  with  objects 
that  can  be  numbered,  before  the  pupils  are  intro- 
duced to  the  use  of  figures,  which  are  the  mere  signs 
of  the  numbers  themselves. 

The  preceding  chart,  Chart  XL,  corresponds  to 
Chart  X.  It  is  simply  the  written  signs  of  the  num- 
bers which  the  children  have  just  learned. 

A  word  as  to  the  use  of  this  chart.  The  children 
are  to  read  the  expressions  of  the  different  numbers 
in  figures,  to  find  the  expression  for  any  number 
which  the  teacher  may  name,  to  find  the  expression  of 
any  number  which  the  teacher  may  show  on  Chart  X., 
and  to  show  upon  Chart  X.  the  number  correspond- 
ing to  any  figures  to  which  the  teacher  may  point. 

If  the  children  understand  into  how  many  tens  and 
units  any  number  may  be  separated,  it  will  generally 
be  sufficient  to  tell  them  that  the  tens  are  written  at 
the  left  and  the  units  at  the  right.  Chart  XI.  may 
now  be  copied  by  the  pupils.  The  teacher  may  now 
have  the  expressions  for  different  numbers  which  he 
finds  on  the  chart  read  and  then  copied. 

To  impress  the  written  expressions  of  the  different 
numbers  from  one  to  one  hundred  upon  the  minds 
of  the  pupils,  the  following  series  may  be  constructed 
and  written  by  the  children  : 

1  +  1-2;  2+1-3;  3  +  i  =  4> 

and  so  on  to  100. 


ADDITION.  67 

17.   ADDITION. 

In  the  two  first  courses,  that  is,  in  the  study  of 
numbers  from  one  to  ten,  and  of  numbers  from  one 
to  twenty,  we  have  recommended  the  simultaneous 
treatment  of  the  four  ground  rules  of  addition,  sub- 
traction, multiplication,  and  division.  At  this  point 
they  should  be  separated.  A  few  words  in  explana- 
tion of  the  reason  will  be  added. 

With  the  size  of  numbers  the  parts  into  which  the 
numbers  can  be  separated  multiply ;  so  that  the  point 
is  soon  reached  where  the  resulting  facts  can  no 
longer  be  impressed  upon  the  memory,  and,  indeed, 
where  this  is  no  longer  necessary.  As  a  rule,  the 
memory  is  to  be  burdened  with  those  facts  only 
which  constitute  the  foundation  upon  which  future 
progress  depends  ;  for  example,  the  addition  and  sub- 
traction tables.  In  teaching  these  we  were  able  to 
ground  all  the  written  exercises  upon  the  direct 
observation  of  the  charts  and  other  objects  ;  but  in 
the  treatment  of  higher  numbers  this  is  impossible. 
But  the  ability  of  the  pupils  at  this  point  has  so 
increased  that  they  are  able  to  solve  a  much  larger 
number  of  problems  in  the  same  time.  We  must  be 
able,  therefore,  especially  where  there  are  several 
divisions  to  be  occupied  at  the  same  time,  to  select 
problems  which  will  be  easy  to  assign,  which  will 
make  the  work  of  correction  easy,  and  which  will 
afford  much  occupation  for  the  pupils. 


68  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

More  than  this,  arithmetic  is  partly  an  art,  and  in 
art  facility  in  doing  presupposes  practice.  Facility, 
however,  can  never  be  attained  unless  the  same  thing 
is  practised  for  a  long  time.  If  a  piano  player  wishes 
to  make  a  movement  absolutely  his  own,  it  is  not 
enough  for  him  to  practise  it  in  its  turn  along  with 
twenty  other  movements ;  he  must  repeat  this  move- 
ment by  itself  over  and  over.  So  facility  in  a  definite 
numerical  operation,  be  it  addition  or  subtraction  or 
any  other,  is  attained  only  through  continued  prac- 
tice in  this  very  operation.  This  practice,  however, 
must  not  be  mere  mechanical  routine,  but  rather, 
thoughtful  practice.  To  secure  this,  careful  work 
must  be  done  in  the  addition  of  units. 

The  instruction  must  begin  with  what  is  easiest, 
and  proceed  gradually  to  the  most  difficult ;  begin 
with  the  addition  of  two,  then  add  three,  and  so  on 
to  nine.  The  exercises  are  to  be  given  at  first  with 
the  help  of  apparatus  for  actual  observation ;  but 
gradually  the  apparatus  is  to  be  dispensed  with,  and 
the  pupils  are  to  be  taught  to  reach  the  required 
results  by  processes  of  thinking.  The  following  may 
serve  as  an  example  of  the  proper  work  in  teaching 
the  addition  of  units  to  tens  or  to  tens  and  units. 
Suppose  the  number  seven  is  to  be  added  to  one  and 
to  the  succeeding  results;  the  steps  would  be  as 
follows  : 

I  +  7  =    8,  which  is  already  known. 
8  +  7  =  15,  may  be  thought  as    8  +  2=10, 
and  10+  5  =  15. 


ADDITION.  69 

15  +  7  =  22,  may  be  thought  as  1 5  +    5  —  20, 

and  20  +  2  =-•  22. 
22  +  7  —  29,  may  be  thought  as  2  +  7  —  9, 

and  20  +  9  =  29. 
29  +  7  =  36,  niay  be  thought  as  29 .  +  I  =  30, 

and  30  +  6  =  36. 
36  +  7  =  43,  may  be  thought  as  36  +  4  =  40, 

and  40  +  3  —  43. 
43  -f-  7  =  50,  may  be  thought  as  3  +  7  =  10, 

and  40+10  =  50. 

SO  +  7-S7. 

57  +  7  =  64,  may  be  thought  as  57  +    3  =  60, 

and  60  +  4  =  64. 
64  +  7  —  71,  may  be  thought  as  64  +  6  —  70, 

and  70+  i  =  71. 
71  +  7  =  78,  may  be  thought  as  7  +  I  =  8, 

and  70+  8_=  78. 
78  +  7  =  85,  may  be  thought  as  78  +  2  =  80, 

and  80+  5  =  85. 
85  +  7  =  92,  may  be  thought  as  85  +  5  =  90, 

and  90  +  2  =  92. 
92  +  7  =  99,  may  be  thought  as  2  +  7  =  9, 

and  90  +    9  =  99. 

If  the  result  falls  within  the  given  ten,  only  the 
units  are  to  be  increased ;  but  if  the  result  reaches 
into  the  next  ten,  the  given  units  are  first  to  be 
increased  to  ten  and  the  remaining  units  added  to 
the  next  ten.  Along  with  this  exercise  in  the  succes- 
sive additions  of  seven,  let  the  corresponding  parts 


70  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

of  the  addition  table  be  carefully  practised.  Write 
upon  the  board  the  numbers  i,  2,  5,  8,  3,  9,  7,  4,  6, 
and  10,  and  have  the  number  7  added  to  each,  until 
the  results  are  perfectly  committed  to  memory.  The 
separating  of  the  number  to  be  added  into  two  parts, 
the  adding  of  the  first  part  to  the  units,  and  the 
adding  of  the  second  part  to  the  next  ten  will  disap- 
pear with  continued  practice  ;  the  addition  of  26  and 
7,  for  example,  will  soon  be  reduced  to  a  single  opera- 
tion, when  the  pupil  is  perfectly  familiar  with  the 
fact  that  6  +  7  =  13. 

Facility  in  addition  grounded  upon  a  thorough 
memorizing  of  the  addition  table  is  the  end  for  which 
the  teacher  should  strive;  but  he  will  not  succeed  in 
having  all  pupils  reach  this  facility  in  the  time  which 
can  properly  be  given  to  the  first  steps  in  addition. 
If  the  teacher  insists  upon  a  perfect  memorizing  of 
the  addition  table  by  the  weakest  pupils,  under  all 
circumstances,  before  proceeding  to  the  addition  of 
larger  numbers,  he  does  a  wrong  to  the  brightest, 
because  he  holds  them  back  upon  the  first  stage  of 
addition  so  long  that  they  become  weary  of  the  work. 
This  is  a  pedagogical  sin,  which  avenges  itself  no 
less  upon  the  individuals  than  upon  the  class.  The 
reason  for  the  unequal  acquisition  of  facility  lies  in 
the  unequal  talent  of  the  pupils  for  impressing  num- 
bers upon  the  memory.  Number  memory  is  not  the 
same  in  all  pupils. 

The  teacher  may,  however,  console   himself  with 


ADDITION.  /I 

the  reflection  that  clear  understanding  and  definite 
comprehension  of  the  process  is  of  more  use  to  the 
student  than  great  facility  in  performing  the  process. 
Every  exercise  must  be  brought  within  the  compre- 
hension of  the  pupil  before  he  is  allowed  to  enter 
upon  a  new  stage  of  work.  The  attainment  of  a 
reasonable  amount  of  facility  is  no  less  desirable  in 
arithmetic  than  in  other  branches  of  study ;  and  yet 
arithmetic  has  this  advantage,  that  the  following 
stages  always  take  up  the  exercises  of  the  preceding, 
and  thus  furnish  an  opportunity  to  increase  the  pupil's 
facility  in  preceding  processes. 

Even  in  private  instruction  it  would  not  be  advisa- 
ble to  keep  a  pupil  of  weak  memory  for  numbers 
upon  the  first  steps  so  long  as  would  be  required  in 
order  to  reach  the  extreme  of  facility.  To  weary  the 
pupil,  to  destroy  his  desire  for  arithmetical  knowl- 
edge, is  an  injury  which  outweighs  any  facility  in 
performing  processes. 

EXAMPLES. 

One  of  the  easiest  ways  of  assigning  examples  for 
practice  at  this  stage  of  the  work  is  to  have  series  of 
numbers  formed,  at  first  by  the  addition  of  the  same 
number,  later  by  the  addition  of  different  numbers 
alternately.  Such  series  occupy  profitably  one  divis- 
ion while  the  teacher  is  busy  with  another.  A  few 
written  figures  will  indicate  the  desired  lesson.  A 
glance  at  the  slate  containing  the  pupil's  work  shows 


72  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

whether  it  is  correct  or  not.     The  following  examples 
will  serve  for  explanation  : 


a. 

b. 

c. 

d. 

0  +  7=  7, 

i  +  7=  8, 

2  +  7=  9, 

4  +  7=  n. 

7  +  7  =  14, 

8  +  7  =  i5, 

9  +  7-16, 

11+7=  1  8. 

14  +  7-21, 

15  +  7-22, 

16  +  7-23, 

18  +  7=  25. 

21  +  7  =  28, 

22  +  7  —  29, 

23  +  7-30, 

25  +  7-  32. 

28  +  7  =  35, 

29  +  7  =  36, 

30  +  7  =  37, 

32  +  7=  39- 

35  +  7-42, 

36  +  7-43, 

37  +  7-44, 

39  +  7-  46. 

42  +  7-49, 

43  +  7  =  50, 

44  +  7-51, 

46  +  7=  53- 

49  +  7-56, 

50  +  7-57, 

51+7-58, 

53  +  7-  60. 

56  +  7-63, 

57  +  7-64, 

58  +  7  =  65, 

60  +  7—  67. 

63  +  7-70,     64  +  7-71,    65  +  7-72,     67  +  7-   74. 

70  +  7  =  77,  7i+7  =  78,  72  +  7  =  79,  74  +  7=  81. 

77  +  7-84,  78  +  7-85,  79  +  7-86,  81+7-88. 

84+7-91,  85  +  7-92,  86  +  7-93,  88  +  7-  95- 

9i+7=98,  92  +  7  =  99,  93  +  7=ioo,  95  +  7-102. 

This  table  represents  the  pupil's  work.  The  prob- 
lems a,  by  c,  and  d  may  be  assigned  by  telling  the 
class  to  add  7  to  o,  i,  2,  and  4  fourteen  times  ;  or 
simply  by  writing  0  +  7,  1+7,  etc. 

Since  the  first  numbers  are  o,  i,  2,  and  4,  the 
results  in  any  horizontal  line  vary  by  i,  2,  and  4,  and 
so  do  the  final  results.  A  glance  at  one  or  two 
places  in  the  vertical  line  and  at  the  end  will  show 
whether  the  work  is  all  right.  The  final  results  will 
be  14  X  7  plus  i,  2,  and  4.  So  any  series  may  be 
dictated,  beginning  with  any  number  from  i  to  9, 


ADDITION.  73 

and  adding  any  number  from    i   to  9,   and  all  the 
results  known  at  a  glance. 

To  give  a  greater  variety,  and  at  the  same  time 
provide  for  reviews,  two  numbers  may  be  added 
alternately,  for  example : 

2  +  4-    6. 

6+3=    9- 

9  +  4=13- 

i3  +  3  =  i6. 

16  +  4--=  20. 

20  +  3  —  23,  etc. 
Compare  this  with  the  series  marked  c  above. 

2  +  7-    9. 
9  +  7  =  16. 
16  +  7  =  23,  etc. 
and  it  is  obvious  that  the  series  will  end  with  100. 

A  word  in  regard  to  written  exercises  in  general. 
While  it  is  true  that  in  the  beginning  of  the  study  of 
numbers  figures  are  a  positive  hindrance,  this  is  by 
no  means  universally  the  case.  Written  exercises 
are  of  the  greatest  importance,  provided  they  are 
properly  connected  with  observation  and  oral  work. 
Practical  life  requires  the  use  of  written  arithmetic 
and  therefore  the  school  must  prepare  the  pupils  for 
it.  But  the  pedagogical  reason  is  still  stronger. 
Children  are  not  all  alike  in  ability.  It  often  happens 
that  in  oral  work  the  brightest  pupils  have  too  little 
to  do  in  proportion  to  their  ability ;  or  that  the  weak- 

OF  THE 


74 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


est  are  behindhand  in  the  solution  of  the  problems, 
so  that  their  real  progress  is  hindered.  Now  written 
exercises,  especially  such  series  as  have  just  been 
recommended,  are  adapted  to  all  conditions  of  the 
class.  Each  can  do  in  a  given  time  what  he  is  able, 
and  all  will  do  good  work.  The  bright  ones  are  not 
kept  back,  and  the  weakest  are  not  overdriven.  If  all 
the  work  is  not  done  by  all  the  pupils,  what  is  done 
is  good  for  all.  Written  work,  then,  is  adapted  to 
all,  while  oral  work  is  often  adapted  only  to  the 
average  talent. 

Moreover,  exercises  in  written  work  for  the  class 

CHART   XII. 

FOR  PRACTICE  IN  THE  GROUND  RULES. 
abcdefghik 


1 

m 

22 
0 

p 
q. 

r 
s 
t 
u 

1 

2 

12 

26 

34 

47 

53 

67 

76 

84 

96 

19 

22 

31 

43 

52 

70 

73 

87 

93 

5 

16 

28 

39 

41 

55 

63 

77 

83 

94 

8 

17 

21 

33 

44 

59 

62 

75 

86 

97 

3 

15 

27 

36 

49 

51 

68 

72 

90 

92 

7 

20 

24 

32 

46 

58 

61 

80 

82 

99 

4 

13 

29 

38 

42 

54 

66 

71 

88 

95 

9 

18 

23 

40 

45 

57 

64 

79 

85 

91 

6 

11 

30 

35 

50 

56 

65 

74 

89 

98 

]0 

14 

25 

37 

48 

60 

69 

78 

81 

100 

SUBTRACTION.  75 

allow  the  teacher  time  to  give  individual  instruction 
to  the  weak  pupils. 

The  preceding  chart,  marked  Chart  XII.,  will  be 
found  very  useful  in  assigning  work  to  be  done  out- 
side the  recitation  hour/ as  well  as  for  exercises,  both 
oral  and  written,  to  be  performed  in  the  class.  Let 
the  numbers  from  one  to  ten  be  added  to  each  of  the 
numbers  and  we  have  one  thousand  examples  in 
addition. 

18.    SUBTRACTION. 

Let  the  numbers  2,  3,  4,  etc.,  to  10,  be  subtracted 
from  100  and  from  the  successive  remainders,  and 
the  exercises  will  be  the  reverse  of  those  explained 
under  addition  ;  for  example  : 

100  —  7  may  be  thought  as  10  —  7  —    3, 

and  loo  —  7  =  93, 
93  —  7  may  be  thought  as  93  —  3  =  90, 

and  90  —  4  =  86, 
86  —  7  may  be  thought  as  86  —  6  =  80, 

and  80  —  i  =  79, 
79  —  7  may  be  thought  as    9  —  7  =    2, 

and  79  —  7  =  72, 
72  —  7  may  be  thought  as  72  —  2  =  70, 

and  70  —  5  =  65  ; 
and  so  on  till  the  remainder  is  less  than  seven. 

If  the  minuend  consists  of  tens  only,  the  subtra- 
hend is  to  be  taken  from  10,  and  the  remainder 
added  to  the  next  ten  below;  for  example  :  100—7 


76  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

becomes  10  —  7— 3,  and  90+3  =  93.  If  the  minuend 
consists  of  tens  and  units,  and  the  units  are  more 
than  the  subtrahend,  the  subtrahend  is  to  be  taken 
from  the  units  and  the  remainder  added  to  the  tens ; 
for  example  :  79—7  is  changed  to  9  —  7  —  2  and 
70  +  2  —  72.  If  the  minuend  consists  of  tens  and 
units,  and  the  units  are  less  than  the  subtrahend,  the 
units  of  the  minuend  are  to  be  subtracted  first,  and 
then  from  the  tens  are  to  be  taken  the  difference 
between  the  units  already  subtracted  and  the  subtra- 
hend ;  for  example  :  93  —  7  is  changed  to  93  —  3  =  90, 
and  90  —  4  —  86. 

This  shows  us  how  important  it  was  to  teach  the 
separation  of  the  numbers  below  10  into  two  parts ; 
and  also  reminds  us  of  the  propriety  of  a  careful 
review  of  the  corresponding  number  before  begin- 
ning a  new  exercise  in  subtraction.  For  example, 
before  giving  exercises  in  the  subtracting  of  seven, 
the  reviews  should  cover  the  following  ground  : 

7  =  6  +  i  or  1+6, 

7-5  +  2  or  2  +  5, 
and  also  7  =  4+3  or  3  +  4; 

10-7,  7-7,  13-7,  ii  -7>  9~7>  iS-7, 
12-7,  14  —  7>  8-7,  16-7,  and  17  -  7. 
In  the  addition  of  9,  the  children  will  often  reach 
the  result  by  adding  10  and  subtracting  i  ;  so  in 
subtracting  9,  they  will  often  reach  the  result  by 
subtracting  10  and  adding  i.  Such  practices  should 


SUBTRACTION.  77 

not  be  allowed  unless  they  are  understood;  which 
will  be  the  case  if  they  are  discovered  by  the  chil- 
dren. But  the  teacher  should  examine  and,  if  neces- 
sary, instruct. 

It  does  not  follow,  however,* that  a  pupil  should  be 
allowed  to  continue  a  practice  in  numerical  computa- 
tion simply  because  he  has  hit  upon  it  and  under- 
stands it.  It  is  usually  better  for  the  teacher  to 
train  the  pupils  in  the  most  expeditious  methods  of 
performing  operations ;  and,  as  a  rule,  one  method  is 
better  than  two.  For  example,  if  the  pupil  is  always 
required  to  reach  results  in  the  addition  and  subtrac- 
tion of  units  in  the  way  described  above,  whenever 
he  is  obliged  to  go  through  a  conscious  process  in 
reaching  the  result,  he  will  reach  the  point  where 
such  operations  are  automatic  much  quicker  than  he 
will  if  he  is  allowed  to  reach  his  results  now  by  one 
process  and  now  by  another. 

Written  exercises  may  be  given  which  are  easy  to 
correct,  consisting  of  series  of  subtractions.  The 
series,  100  —  7,  etc.,  will  end  with  2  ;  for  7  in  100  = 
14,  and  2  remainder.  The  series  99  —  8  ends  with  3 ; 
since  99-^8  =  12,  and  3  remainder.  Thus  exercises 
may  be  set  consisting  of  a  dozen  or  more  subtrac- 
tions, so  constructed  that  a  glance  at  the  final  result 
will  show  whether  the  work  is  correct. 

Chart  XII.  may  be  made  helpful  in  oral  drill, 
since  i,  2,  3,  etc.,  may  be  subtracted  from  each 
number ;  and  thus  the  labor  of  the  teacher  may  be 
materially  lessened. 


78  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

19.   CONNECTED  ADDITION  AND  SUBTRACTION. 

For  the  sake  of  variety,  as  well  as  for  the  purpose 
of  reviewing  the  work  of  addition,  series  of  alternate 
additions  and  subtractions  may  be  given  for  written 
work ;  for  example  : 

100-7-93, 
93+4-97,  100-3-97; 

97-7  =  9°> 

90  +  4-94,  97  -  3  -  94 ; 

94-7=87, 

87  +  4-91,  94-3=9i; 

and  so  on.  Since  subtracting  7  and  adding  4  reduces 
a  number  3,  every  second  result  must  be  the  same  as 
if  3  were  subtracted. 

i+7=    8, 

8-3-    5>  1+4-5- 

5 +  7- 12, 
12-3-9,  5+4  =  9>  etc- 

Since  adding  7  and  subtracting  3  increases  a  num- 
ber 4,  every  second  result  must  be  the  same  as  if  4 
were  added. 

Work  of  this  kind  is  easy  to  assign  and  easy  to 
examine.  The  above  illustrations  are  designed  merely 
as  suggestions  of  what  may  be  done. 

So  far  in  our  treatment  of  numbers  consisting  of 
two  places  we  have  added  and  subtracted  only  units ; 
but  it  would  be  a  good  preparation  for  work  with 


CONNE  C  TED  ADDITION  AND  S  UB  TRA  C  TION.     79 


numbers  from  i  to  1,000  to  add  and  subtract  num- 
bers larger  than  10  at  this  stage.  The  following  is 
suggested  as  a  good  order  of  work  : 


a.  Tens  to  tens. 

20  +  30, 

50+  10, 
10  +  20,  etc. 

b.  Tens  to  tens  and  units. 

36+40, 

30  +  40  =  70  ;  so 

36  +  40  =  76. 

c.  Tens  and  units  to  tens. 

30  +  25, 

30  +  20  -  50, 

50+    5  =  5$. 


a.  Tens  from  tens 

100—  10, 
100—  30, 
90  —  70,  etc. 

b.  Tens  from  tens  and  units. 

96  -  40, 

90  —  40  =  50  ;  so 

96  —  40--=  56. 

c.  Tens  and  units  from  tens. 

90  -  63, 

90  -  60  =--  30, 

30-    3  =  27. 


d.  Tens  and  units  to  tens  and  units. 
32  +  44  —  32  +  40  +  4, 
68  +  28  =  68  +  20  +  8. 

d.  Tens  and  units  from  tens  and  units. 
96  -  34  =  96  —  30  -  4, 
83—65  -83—60-5. 

These  exercises,  at  least  those  marked  a,  b,  and  c, 
should  be  readily  performed  by  the  pupils  orally 
before  they  are  changed  to  written  exercises.  In 
the  written  work,  a  union  of  addition  and  subtraction 
may  take  place  in  the  same  series  of  exercises,  as 
shown  below : 


80  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

1.     IO+2O  =  3O.       2.     10+30  =  40.       3.  100  —  30  =  70. 

20  +  20  =  40.  20+30=50.  90  —  30  =  60. 


etc. 


4.  IOO  —  2O= 


+  10 

IO  +  5O 
-40 
4  +  40 

—  2O 
98—40 

+  20 
10  +  25 

—  20 
19.  100  —  25 

+  20 
22.      12  +  24 

—  12 


10. 


13. 


16. 


5. 


11- 


14. 


17. 


6. 


12- 


15. 


18. 


etc. 
IO  +  4O 

—20 

8.  IOO—4O 
+20 

7+50 
—30 
96—50 
+30 
10  +  46 
—30 

20.  100  —  46 
+30 

23.         7  +  36 
-18 

Similar  series  of  numbers  may  be  given  indefinitely 
as  the  needs  of  the  class  require. 

The  following  rules  may  be  useful  : 

To  add  one  number  between  10  and  100  to  an- 
other, add  first  the  tens  and  then  the  units  ;  for 
example:  57  +  39;  S7  +  30=87;  8^+9  =  96. 

To  subtract  one  number  between  10  and  100  from 
another,  subtract  first  the  tens  and  then  the  units  ; 
for  example:  77~49;  77—  4O  =  37;  37  -9^28. 

A  teacher  ought  to   be    satisfied  with    the  weak 


etc. 
IO+5O 

—30 

9.  IOO  —  5O 
+30 

9  +  50 
—40 
97—50 
+40 
10+38 
—20 

21.   100—28 
+20 

24.        1+28 
—14 


CONNECTED  ADDITION  AND  SUBTRACTION.     8 1 

pupils  if  they  can  solve  problems  in  these  ways,  and 
not  try  to  teach  them  shorter  processes.  It  is  better 
for  a  pupil  to  be  certain  in  one  way  than  to  be  uncer- 
tain in  several. 

It  was  previously  shown  that  there  was  great 
advantage  in  being  able  to  increase  any  fundamental 
number  to  10;  there  is  a  like  advantage  in  being 
able  to  increase  any  number  below  a  hundred  to  a 
hundred.  It  is  well,  therefore,  to  drill  the  pupils  in 
such  exercises  as  these  : 

86  and  how  many  are  100  ? 

86  +  4  =  90;     90+10=100;     hence  14. 

67  and  how  many  are  100? 

67  +  3  =  7o;     70+30=100;     hence  33. 
48  and  how  many  are  100? 

48  +  2  =  50;     50+50=100;     hence  52. 

Chart  XII.  affords  abundant  matter  for  drill  in 
the  addition  of  numbers  below  100 ;  for  example,  in 
adding  48  to  each  number  on  the  chart,  there  are 
100  additions.  But  each  other  number  below  100 
may  be  added ;  which  makes  5,000  examples  in  addi- 
tion. Or,  by  how  many  does  53  differ  from  each 
number  on  Chart  XII.  ?  In  answering  this,  the 
child  performs  100  subtractions.  But  the  same  may 
be  asked  of  all  the  other  numbers  below  100;  which 
gives  5,000  examples  in  subtraction.  Add  24  to  each 
number  in  the  first  five  vertical  columns ;  in  the  first 
four  horizontal  lines,  etc.  Remember  that  practice 
makes  perfect. 


82  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

2O.    MULTIPLICATION. 

The  multiplication  table  is  the  foundation  of  the 
process  of  multiplication.  It  is  the  tools  without 
which  neither  multiplication  nor  division  can  be  per- 
formed. Hence  the  child  must  make  it  so  completely 
his  own  that  it  cannot  be  forgotten  and  that  it  will 
always  be  present  to  him  in  the  twinkling  of  an  eye 
when  it  is  needed  for  use.  It  is  in  the  power  of  the 
teacher  to  render  such  help  to  the  little  ones  as  will 
spare  the  tears  which,  without  such  help,  will  be 
sure  to  flow  when  the  demand  is  made  upon  them  to 
learn  the  multiplication  table  by  heart. 

If  the  teacher  wishes  happily  to  avoid  these  break- 
ers he  must  be  sure  that  two  things  always  exist  in 
proper  relation  one  to  the  other,  —  intelligence  and 
practice.  Intelligence,  which  is  gained  only  through 
direct  observation,  was  formerly  neglected ;  but  the 
tendency  at  the  present  time  is  to  neglect  the  prac- 
tice. Instead  of  drilling  the  pupils  thoroughly  in  the 
multiplication  table  in  school,  by  means  of  recitations 
and  questioning,  and  by  means  of  connecting  the 
work  at  every  step  with  the  preceding  lessons,  many 
teachers  are  satisfied  with  making  the  children  under- 
stand how  it  is  formed,  and  leaving  the  memorizing 
to  be  done  as  home  lessons.  But  it  is  the  special 
task  of  the  teacher  to  show  the  children  how  they 
should  learn.  In  order  to  point  out  to  inexperienced 
teachers  what  exercises  they  may  introduce  to  advan- 


TEACHING    THE  MULTIPLICATION   TABLE.       83 

tage  while  the  pupils  are  committing  the  multiplica- 
tion table  to  memory,  it  will  be  necessary  to  go 
somewhat  into  details. 

Since  thorough  drill  requires  a  long  time,  it  is 
recommended  to  make  a  preparation  for  the  learning 
of  the  multiplication  table  while  teaching  addition 
and  subtraction.  When  the  pupils  have  thoroughly 
learned  to  add  and  subtract  the  number  two,  they 
may  be  taught  to  multiply  by  two ;  when  they  have 
learned  to  add  and  subtract  the  number  three,  they 
may  learn  the  threes  of  the  multiplication  table,  etc. 
By  this  course  sufficient  time  may  be  secured  for 
reviews,  which  here  are  indispensably  necessary, 
since  upon  them  depends  the  impressing  of  numbers 
upon  the  memory. 

21.   TEACHING  THE  MULTIPLICATION  TABLE. 

TWOS   OF   THE   MULTIPLICATION   TABLE. 

In  our  treatment  of  numbers  from  i  to  20  we  have 
already  found  once  2,  2  times  2,  and  so  on  to  10 
times  2,  and  we  will  rejoice  at  whatever  has  remained 
in  the  memory ;  still  it  is  necessary  to  develop  the 
facts  again. 

Place  two  balls  on  the  numeral  frame,  or  two 
points  on  the  board  beside  each  other,  thus  : 

•        • 

and  ask,  How  many  balls  are  there  ?  Then  put  two 
more  balls  with  them,  thus  : 


84  ARITHMETIC  IN  PRIMARY  SCHOOLS. 


and  ask,  How  many  times  two  balls  are  there  ?  How 
many  are  two  times  two  balls  ?  How  many  are  two 
times  two  ?  How  many  are  two  twos  ? 

Just  so  may  the  ideas  of  3,  4,  5,  6,  7,  8,  9,  and  10 
times  2  be  developed.  In  doing  this  the  following 
figure  will  be  formed,  and  the  following  expressions 
of  the  truths  which  it  represents  should  be  repeated 
many  times,  both  by  individuals  and  in  concert : 


•     •       Once 

2    is 

2. 

10  times 

2 

are  20. 

•     •       2  times 

2  are 

4- 

9      " 

2 

"    18. 

•    •       3  '  " 

2     " 

6. 

8      " 

2 

"    1  6. 

•    •       4      " 

2     " 

8. 

7     " 

2 

M     14. 

•    •      5     « 

2     " 

IO. 

6     " 

2 

"      12. 

•     •       6      " 

2     " 

12. 

5      " 

2 

"     IO. 

•    •       7     " 

2     " 

14. 

4      " 

2 

"      8. 

•    •       8      " 

2     " 

1  6. 

3      " 

2 

"      6. 

.    .       9     « 

2     " 

1  8. 

2        " 

2 

"      4- 

•    •     10     " 

2     " 

20. 

Once 

2 

is     2. 

It  will  help  if  the  children  are  led  to  find  that,  for 
example,  3  times  2  units  are  just  as  many  units  as  2 
times  3  units.  Thus,  in  the  following  figure  there 
are  3  rows  of  2  points  each,  and  there  are  also  2  rows, 
of  3  points  each. 


TEACHING    THE  MULTIPLICATION   TABLE.      85 

So  it  may  be  shown  that 

4  times  2  are    8,  and  2  times    4  are    8. 

5  "     2   "    10,    "    2     "        5    "    10. 

6  "       2     "      12,      "     2       "  6     "      12. 

7  "     2   "    14,    "    2     "        7   "    14. 

8  "     2    "    1 6,    "    2     "        8    "    1 6. 

9  "     2    "    1 8,    "    2     "        9   "    1 8. 

IO        "        2     "      2O,      "      2       "          IO     "      2O. 

The  results  stated  at  the  right  are  already  known 
as  the  sums  of  equal  numbers.  The  one  set  of  state- 
ments assists  the  pupil  in  remembering  the  other, 
yet  the  truths  ought  not  to  be  confused.  Practical 
examples  like  the  following  will  guard  against  such 
confusion  : 

A  mother  gave  her  son  4  apples  yesterday  and  4 
to-day ;  how  many  times  did  he  receive  4  apples  ? 
How  many  are  2  times  4?  How  many  times  4  is  8? 
How  many  fours  can  be  made  of  8? 

A  woman  gives  her  child  2  apples  daily ;  how 
many  times  2  apples  does  he  receive  in  4  days  ? 
How  many  are  4  times  2  ?  How  many  times  2  is  8  ? 
How  many  twos  can  be  made  of  8  ? 

Let  the  children  illustrate  the  multiplication  table, 
as  shown  above,  with  points  on  their  slates,  and  affix 
the  results,  thus  : 

•  •         4, 

•  •         6,  etc. 

If  they  are  to   study  the  multiplication  table  at 


86  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

home,  let  them  first  construct  it ;  otherwise  it  is  apt 
to  have  little  meaning.  The  impressing  of  the  facts 
out  of  their  order  is  to  be  effected  mainly  through 
question  and  answer,  and  is  proper  work  for  the 
schoolroom.  The  more  varied  the  practice,  however, 
the  more  firmly  the  facts  are  impressed  upon  the 
memory.  Hence  the  reverse  form  of  viewing  the 
facts  is  to  be  used.  For  example  :  How  many  times 
is  2  in  12?  How  many  twos  in  12?  How  often  is 
2  contained  in  12?  How  often  can  we  take  2  from 
12?  etc.  It  is  well  to  spend  a  week  or  two  on  the 
number  2 ;  and  a  portion  of  each  lesson  should  be 
given  to  practical  applications  ;  for  example  : 

1  whole    =  2  halves,      i  apple  costs  2  cts. 

2  wholes  —  4  halves.     2  apples  cost  4  cts. 

3  «       =  6      "  3      "         "     6    " 
etc.                 etc.  etc.  etc. 

THREES   OF  THE   MULTIPLICATION   TABLE. 

•  •  •  Let  the  course  of  instruction  be  as  fol- 

•  •  •  lows : 

•  •  •  a.  Construct  the  table  on  the  frame  or 

•  •  •  board,  as  in  the  margin. 

•  •  •  b.  Practise  alone  and  in  concert  forwards. 

•  •  •  c.  Practise   alone   and   in    concert   back- 

•  •  •  wards. 

•  •  •  d.  Question  out  of  the  regular  order. 

•  •  •  e.  Let  the  children  make  the  same  on 

•  •  •  their  slates. 


TEACHING    THE  MULTIPLICATION   TABLE.      8/ 

The  results,  i,  2,  3,  4,  5,  6,  and  10  times  3,  the 
children  will  easily  retain  ;  for  3  X  2  has  been  already 
learned  in  studying  the  twos ;  3  X  3,  in  studying  the 
number  picture  for  9  ;  4X3,  in  the  treatment  of  12  ; 
5  X  3,  in  the  study  of  1 5  on  Chart  VI. ;  6  X  3,  in  the 
study  of  1 8  on  Chart  VI.  ;  and  10X3  =  3X10  =  3 
tens.  These  results  will  now  afford  little  difficulty ; 
7,  8,  and  9  times  3  will  cause  more.  But  9X3  — 
10x3-1x3;  7><3  =  7+7  +  7;  8x3-8  +  8  +  8; 
and  all  these  are  to  be  taught  from  rows  of  points. 

The  following  applications  are  suggested : 

1  orange  costs  3  cents, 

2  oranges  cost  6  cents, 

3  oranges  cost  9  cents,  etc. 

FOURS  OF  THE   MULTIPLICATION   TABLE. 

The  course  of  exercises  is  the  same  as  in  teaching 
the  twos  and  threes.  More  or  less  are  already  known 
of  i,  2,  3,  4,  5,  and  10X4;  so  fix  these  numbers 
first.  Connect  9X4  with  10X4;  6X4  with  the 
known  5x4.  Take  special  pains  with  7X4  and 
8X4.  Apply  as  follows  : 

1  horse  has  4  legs, 

2  horses  have  8  legs, 

3  horses  have  12  legs,  etc. 

FIVES   OF   THE   MULTIPLICATION   TABLE. 

a.  The  pupil  knows  i,  2,  3,  4,  5,  10  X  5. 

b.  He  learns  5  X  5  =  25  easily  from  the  sound. 


88  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

C.    9  X  5  =  10  X  5  —  I  X  5. 

d.  6x5  —  5x5  and  5  =  30. 

e.  8x5  =  2  times  4X5  =  2  times  20,  or  4  times 
2x5=  4  times  10. 

f.  7  x  5  is  to  be  connected  with  6x5. 

Always  direct  the  practice  so  as  to  connect  the 
thing  to  be  learned  with  what  precedes ;  first  a,  then 
by  then  a  and  b ;  then  c,  then  ay  by  and  c ;  then  d, 
then  ay  by  c,  d\  then  ey  then  #,  by  cy  dy  ey  etc.  Apply 
thus  :  i  five-cent  piece  =  6  cents,  etc. 

SIXES   OF  THE   MULTIPLICATION   TABLE. 

a.  i,  2,  3,  4,  5,  and  10  X  6  are  known. 

b.  6  X  6  is  remembered  by  the  sound. 

c.  9  X  6  is  to  be  connected  with  10  X  6. 

d.  7  X  6  is  to  be  connected  with  6x6. 
*?.    8  X  6  demands  special  work. 
Application  :   i  week  has  6  working-days,  etc. 

SEVENS    OF  THE   MULTIPLICATION   TABLE. 

a.  i,  2,  3,  4,  5,  6,  and  10  X  7  are  known. 

b.  7  X  7  is  easy  to  remember  from  the  sound. 

c.  9  X  7  is  to  be  connected  with  10x7. 

d.  8  X  7  is  to  be  connected  with  7x7. 
Application  :   i  week  has  7  days,  etc. 

EIGHTS   OF  THE   MULTIPLICATION   TABLE. 

a.  i,  2,  3,  4,  5,  6,  7,  10  X  8  are  known. 

b.  8x8  is  easy  to  learn  from  the  sound. 


TEACHING    THE  MULTIPLICATION  TABLE.      89 

c.  9X8  —  10X8  —  1X8. 
Application :  8  boys  sit  in  i  row,  etc. 

NINES   OF  THE   MULTIPLICATION   TABLE. 

a.  i,  2,  3,  4,  5,  6,  7,  8,  10  x  9  are  known. 

b.  9  X  9  is  learned  from  the  sound,  also  connected 
with  10  x  9. 

Application :   i  yard  costs  9  cents,  etc. 

TENS    OF   THE   MULTIPLICATION   TABLE. 

The  result  is  already  known. 

Application  :   i  dime  is  worth  10  cents,  etc. 

If  the  children  are  made  to  observe,  to  recall,  and 
to  connect  the  unknown  with  the  known,  in  the  way 
just  pointed  out,  they  may  soon  be  brought  to  under- 
stand any  part  of  the  multiplication  table.  But  the 
teacher  must  discriminate  sharply  between  under- 
standing and  knowing.  Knowing  presupposes  con- 
tinued practice  and  diligent  repetition  of  what  pre- 
cedes ;  hence  the  pupil  should  never  pass  to  a  new 
sentence  without  reviewing  what  goes  before.  Let 
the  new  sentence  be  a  reward  for  what  is  already 
learned,  so  that  the  children  will  be  accustomed  to 
find  the  reward  for  learning  in  the  act  of  learning. 

Knowing  the  multiplication  table  implies  readiness 
for  use.  The  child  must  remember  only  the  result, 
not  the  process  of  reaching  it.  Question  must  follow 
answer  instantly.  In  a  word  the  multiplication  table 


90  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

must  be  absolutely  a  thing  of  the  memory.  On 
thought  of  the  words  seven  times  jive  the  thought  of 
the  word  thirty-five  must  instantly  follow.  Perfect 
understanding  comes  through  illustration ;  perfect 
memorizing,  through  diligent  use  and  through  fre- 
quent repetition  almost  endlessly  continued. 

What  we  have  already  explained  is  only  a  prepa- 
ration for  learning  the  multiplication  table.  This 
preparation  was  aimed  at  in  the  addition  of  numbers 
from  one  to  a  hundred  ;  for  example,  when  the  pupil 
was  exercised  in  the  successive  additions  of  six, 
ground  was  broken  for  learning  the  sixes  of  the  table. 
The  results  can  be  fixed  in  the  mind  only  through 
continuous  application.  All  up  to  this  point  is  only 
a  preparation  for  learning  the  table  in  its  written 
form. 

22.   APPLYING  THE  TABLE  TO  WRITTEN  WORK. 

The  written  sign  for  multiplication  is  an  inclined 
cross,  thus,  X,  and  means  time  or  times. 

If  we  should  write  down  the  numbers  from  I  to 
10,  and  ask  the  pupils  to  use  them  successively  as 
multipliers  of  a  given  number,  we  should  by  this 
means  assist  them  to  reach  the  results  by  the  suc- 
cessive additions  of  the  number  to  itself.  However 
necessary  this  order  may  be  in  the  development  of 
the  table  and  for  its  thorough  comprehension,  still  a 
practical  mastering  of  the  same,  a  ready  working 
knowledge  of  it,  demands  its  application  out  of  this, 
order. 


APPLYING   THE    TABLE  TO  WRITTEN  WORK.    9! 

In  order  to  fix  in  the  minds  of  pupils  that  they  are 
always  working  for  results,  and  not  merely  for  prac- 
tice, write  down  the  numbers  from  i  to  10  in  the 
following  order  : 

i,  2,  5,  8,  3,  7,  4,  9,  6,  10. 

Now,  partly  for  the  purpose  of  introducing  variety 
into  the  work,  and  partly  for  the  sake  of  review,  con- 
nect both  addition  and  subtraction  with  exercises  in 
multiplication.  The  beginning  of  the  work  of  mul- 
tiplying with  two  may  be  as  follows  : 

1X2+1=3.   8x2  —  1  =  15.    9X2+1=  . 

1X2—1=  I.    3X2+1=   .     9X2—1=   . 

2x2+1=5.   3x2—1=  .    6x2  +  1=  . 
2x2—1=3.   7x2  +  1=  .    6x2—1=  . 

5X2+1  =  11.         7X2—1=       .          10X^2+1=      . 
5X2—  I=:     9.         4X2  +  1=       .          10X2—1  = 
8X2+1  =  17.         4X2—1=       . 

Substitute  the  numbers  2,  3,  etc.,  to  10,  for  i  in 
the  above  exercises,  as  the  numbers  to  be  added  and 
subtracted,  and  you  have  200  examples  in  multiplica- 
tion by  2.  Now  substitute  the  numbers  3,  4,  5,  etc., 
to  10,  in  place  of  2  in  the  above  examples,  as  the 
numbers  to  be  multiplied,  and  you  have  1,800  exam- 
ples in  multiplication.  With  one-half  of  these  are 
connected  examples  in  addition,  and  with  the  other 
half  examples  in  subtraction. 

In  assigning  work  of  this  kind  it  is  only  necessary 
for  the  teacher  to  write  or  dictate  one  or  two  exam- 


92 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


pies  of  a  kind ;  for  the  pupils  can  readily  invent  the 
rest  of  the  series  up  to  20  examples. 

When  the  pupils  are  familiar  with  the  multiplica- 
tion of  whole  numbers,  as  indicated  above,  the  mul- 
tiplication of  fractions  may  be  introduced  with  profit. 
If  the  treatment  of  fractions  is  to  be  easy  and  pleas- 
urable, the  pupils  must  be  made  entirely  familiar 

*• 
f 

i- 


54 

54 

54 

54 

54 

54 

. 
54 

54 

% 

54 

1/2 

54 

54 

54 

2  = 

4  = 

5 
6 

T- 

8: 

9 

IO  - 


=  ¥• 
=  ¥• 

=  ¥• 
=¥• 


with  fractions  themselves,  as  well  as  with  the  mode 
of  expressing  them  ;  and  for  this  purpose  the  appli- 
cation of  the  multiplication  table  furnishes  an  excel- 
lent opportunity.  We  will  begin  with  the  represen- 
tation and  multiplication  of  halves. 

That   one   whole   is   equal   to   two   halves   may  be 
illustrated  by  the  actual  division  of  a  piece  of  paper, 


APPLYING   THE   TABLE  TO  WRITTEN  WORK.     93 

an  apple,  etc.,  into  two  equal  parts.  Then  the  same 
may  be  illustrated  by  dividing  a  line  or  a  circle.  As 
a  result  of  the  treatment  of  lines  and  circles  in  this 
way,  the  preceding  work  will  appear  on  the  board 
and  on  the  pupils'  slates. 

It  is  only  necessary  to  tell  the  pupils  that  half  is 
written  thus,  T,  and  that  the  number  of  halves  is 
shown  by  the  figure  above  the  line.  The  method  of 
writing  fractions  needs  much  practice  on  the  slates. 

When  the  children  have  become  familiar  with  the 
meaning  and  representation  of  halves,  they  may  per- 
form the  following  series  of  examples  : 

i+l  =  f.  8-|  =  9  +  1  = 

1-1  =  1-  3  +  1=  9-1  = 


2 

+  l  =  f. 

3 

-*- 

6  + 

l 

2 

-i=* 

7 

+  i  — 

6- 

1 

5 

+  ^  = 

7 

-i  = 

10  + 

1 

S 

-1  = 

4 

+  ^  = 

10- 

-?r 

8  +  1-  4-^1  = 

These  exercises  may  be  increased  to  almost  any 
extent  by  adding  and  subtracting  more  than  1. 

A  preparation  for  division  may  be  made  by  orally 
questioning  the  children  in  this  way :  How  many 
whole  ones  in  --/-  ?  How  many  whole  ones  and 
halves  in  -^-P 

In  a  similar  manner  may  thirds,  fourths,  etc.,  to 
tenths,  be  treated. 


94 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


23.    CONSTRUCTING  THE  TABLE. 

The  practice  of  beginning  the  work  in  multiplica- 
tion by  committing  to  memory  a  ready-made  multi- 
plication table  cannot  be  too  strongly  condemned. 
But  if  the  pupil  writes  down  the  facts  in  tabular  form 
as  fast  as  he  learns  them,  he  will  construct  for  him- 
self the  following  table,  designated  as  Chart  XIII. 
This  will  not  only  serve  to  recall  the  facts,  but  will, 
at  the  same  time,  be  a  means  of  teaching  the  facts 
intuitively. 

CHART    XIII. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

That  five  times  four  are  twenty  may  be  shown 
thus  :  Count  down  the  chart  at  the  left  to  4 ;  there 
are  four  rectangles  ;  at  the  right  of  these  are  four ; 


CONSTRUCTING    THE    TABLE.  95 

and  so  on  to  the  row  beginning  with  5  in  the  upper 
row.  That  is,  5  fours  are  20.  If  now  we  count  the 
rectangles  along  the  top  row  from  i  to  5,  we  find  five 
in  the  row  ;  below  these  is  another  row  of  five  ;  and 
so  on  to  the  row  beginning  with  4  in  the  left-hand 
column.  That  is,  four  fives  are  20. 

In  this  way  we  obtain  an  intuitive  knowledge  that 
5  fours  are  20  and  that  4  fives  are  20.  In  the  same 
way  all  the  facts  of  the  multiplication  table  may 
be  demonstrated.  Such  demonstration  will  lay  the 
foundation  for  the  fact,  to  be  learned  by  and  by,  that 
the  product  is  not  affected  by  the  order  of  the 
factors. 

This  table  is  well  fitted  to  teach  the  resolution  of 
numbers  into  their  factors  ;  for  if  the  children  know 

that 


6  x  4  =  24, 

They 

24  -  6  X  4  ; 

4  x  6  =  24, 

must 

24  -  4  X  6  ; 

3  x  8  -  24, 

also 

24  =  3  X  8  ; 

8  x  3  =  24, 

know  that 

24  =  8  X  3  ; 

and  every  fact  in  multiplication  should  be  followed 
by  the  corresponding  fact  in  factoring.  This  is  an 
excellent  preparation  for  division.  So  also  is  the 
changing  of  fractions  to  whole  numbers  ;  for  exam- 
ple :  f  =  4andi;  y  =  5  and  f. 

In  order  to  prepare  the  pupils  for  the  work  of 
multiplication  when  dealing  with  larger  numbers, 
they  should  here  be  taught  to  multiply  numbers 
consisting  of  tens  and  units.  The  following  are 
illustrations  of  the  work  : 


96  ARITHMETIC  IN  PRIMARY  SCHOOLS. 


3  x  24  -      . 
3  x  20  =  60. 
3><    4==i2. 

4  x  18  = 
4  x  10  =  40. 
4  x    8-32. 

8X12=       . 

8x10  =  80. 
8  x    2  =  16. 

3  x  24  =  72.  4  x  18  =  72.  8x12=  96. 

The  written  work  in  multiplication  at  this  stage  is 
limited.  We  can,  however,  use  the  following  series 
of  numbers  from  Chart  XII.  : 

a,  by  c,  d  by  2  ;  a,  b  by  5  ;  a  by    8  ; 

a,  b,  c       by  3  ;  a,  b  by  6  ;  0  by    9  ; 

a,  b  by  4 ;  #      by  7  ;  #  by  10. 

In  what  precedes  we  have  shown  how,  through 
objective  illustrations,  the  products  of  numbers  2,  3, 
etc.,  to  10,  may  be  understood  by  children,  and  how 
these  products  may  be  fixed  in  their  minds  by  oral 
and  written  exercises.  These  products  form  the 
so-called  multiplication  table,  by  the  help  of  which 
many  arithmetical  operations,  which  might  be  per- 
formed by  the  repeated  addition  of  the  same  number, 
may  be  materially  shortened.  While  constructing 
this  table  the  pupil  has  found  that  the  multiplication 
of  a  number  is  finding  the  sum  obtained  by  additions 
of  the  same  number.  He  has  himself  found  the 
product  by  the  addition  of  the  same  number ;  and 
he  can  in  the  same  way  find  it  again,  should  it  escape 
his  memory.  But  facility  in  computation  requires 
that  these  products  be  made  things  of  the  memory. 
Remembering  how  to  find  a  product  is  to  be  distin- 
guished, from  remembering  the  product  itself.  In  a 


;   B   N 
PREPARATION  FOR  DIVISION.  97 

subject  like  arithmetic,  where  the  understanding  is 
constantly  called  into  exercise,  there  must  be  no 
halting  of  the  memory.  Hence  we  seek  to  make  the 
facts  of  the  multiplication  table  so  appropriated  by 
the  mind  that  they  will  seem  to  be  the  necessary 
qualities  of  the  memory  itself.  This  is  to  be  accom- 
plished through  continuous  practice  in  computation, 
provision  for  which  has  been  made  in  what  precedes. 


24.    PREPARATION  FOR  DIVISION. 

It  is  well  so  to  treat  the  subject  of  multiplication 
as  to  prepare  the  pupils  for  division.  We  have  been 
finding  products  when  we  knew  the  factors ;  but  the 
process  is  to  be  reversed,  and  we  are  to  find  the 
factors  when  the  product  is  given ;  or,  we  are  to  find 
one  factor  when  the  product  and  the  other  factor  is 
given.  To  divide  24  objects,  beans,  sticks,  etc.,  into 
4  equal  parts,  put  first  one  object  in  each  of  4  differ- 
ent places,  then  distribute  4  more  in  the  same  way, 
then  4  more,  and  so  on  till  the  24  are  all  distributed. 
We  now  have  6  objects  in  each  place.  One  of  the 
four  parts,  which  together  contain  24  objects,  con- 
tains 6  objects.  It  follows  that  24  is  4  times  6,  and 
also  that  the  fourth  part,  or  \,  of  24  is  6. 

This  finding  of  the  second  factor  is  accomplished 
through  successive  subtractions  of  the  same  number ; 
but  facility  in  reckoning  requires  the  pupil  to  be 
able,  given  the  product  and  one  factor,  to  know  the 


98  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

other  instantly.  He  must  be  taught .  to  perform  a 
process  the  opposite  to  what  is  required  in  finding 
the  product  from  the  factors.  He  must  be  able  to 
tell  at  once  how  large  a  certain  part  of  a  number 
is,  how  often  a  certain  number  can  be  taken  from 
another,  or  how  often  a  certain  number  is  contained 
in  another. 

Much  may  be  done  in  connection  with  multiplica- 
tion to  prepare  the  student  for  such  work.  The 
following  figure 


•  •          • 

•  99 


Illustrates  these  truths  : 

4  times    6  =  24 ;  6  times    4  =  24 ; 

\    of      24=-    6;  i    of      24=    4; 

4    in       24  =    6  times ;  6    in       24—4  times. 

Therefore,  to  the  usual  questions,  How  many  are 
4  times  6  ?  etc.,  add  :  From  what  number  can  4  sixes 
be  taken  ?  6  fours  ?  From  what  number  can  4  be 
taken  6  times  ?  Six  4  times  ?  In  what  number  is  4 
contained  6  times  ?  Six  4  times  ?  What  is  the  fourth 
part  of  24  ?  The  sixth  ?  Of  what  number  is  6  the 
fourth  part  ?  Four  the  sixth  part  ? 

If  4  apples  cost  24  cents,  how  much  will  i  cost  ? 
If  6  cost  24  cents,  what  costs  i  ?  Charles  stands  24 
soldiers  in  4  rows  ;  how  many  stand  in  i  row  ?  What 


DIVISION.  99 

part  of  24  is  one  row?    24  is  how  many  times  6? 
How  many  times  4  ?  etc. 

If  such  questions  as  these  are  asked  in  connection 
with  the  development  and  application  of  the  multipli- 
cation table,  a  good  preparation  will  be  made  for  the 
next  stage  of  the  work,  namely,  division. 

25.    DIVISION. 

There  are  two  kinds  of  division,  namely,  separating 
a  number  into  equal  parts,  and  finding  how  often  one 
number  is  contained  in  another.  As  an  example  of 
the  first  kind,  suppose  6  children  have  48  cents,  and 
the  question  is,  How  many  cents  will  each  child 
have,  if  the  cents  are  equally  divided  among  them  ? 
We  reason  that  each  child  will  have  one-sixth  of  48 
cents,  or  8  cents.  Here  is  an  actual  division,  a  sepa- 
ration of  the  48  cents  into  6  equal  parts. 

As  an  example  of  the  second  kind  of  division,  let 
the  question  be,  Among  how  many  children  can  48 
cents  be  divided  if  each  child  receives  6  cents  ?  We 
reason  thus  :  From  48  cents  6  cents  apiece  can  be 
given  to  as  many  children  as  the  times  that  6  cents 
can  be  taken  from  48  cents,  or  the  times  that  6  cents 
are  contained  in  48  cents,  namely,  8  times ;  hence, 
among  8  children.  Here  we  have  found  how  many 
times  6  cents  are  contained  in  48  cents. 

In  both  of  these  examples  the  number  48  is  divided 
into  6  equal  parts  ;  but  while  the  answer  to  the  first 


100          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

question  is  8  cents,  the  answer  to  the.  second  is  8 
children.  From  these  examples  it  appears  that  the 
solution  should  always  correspond  to  the  question. 
A  confusion  of  the  ideas  involved  in  these  two  pro- 
cesses is  a  sign  of  a  thoughtless  solution.  The 
teacher  should  guard  against  this  confusion  from  the 
first,  and  never  allow  such  solutions  as  the  following : 

1.  If  48  cents  are  divided  equally  among  6  chil- 
dren, each  child  will  receive  as  many  cents  as  6  is 
contained  in  48.     Six  children  are  not  contained  in 
48  cents.     Here  6,  that  is,  6  cents,  is  contained  in 
48,  that  is,  48  cents,  8  times,  and  not  8  cents  ;   and 
the    comparison    is    really   between   the    number   of 
cents  and  the  number  of  times  that  48  contains  6. 
A  better  solution  would  be  this  :    Each  child  would 
receive  one-sixth  of  48  cents,  or  8  cents. 

2.  Among  how   many   children   can   48   cents   be 
divided,  if  each  child  receives  6  cents  ?     One-sixth 
part  of  48  is  8  ;    therefore,  8  children.     But  48  was 
48  cents,  and  not  48  children.      A   better  solution 
would  be  this  :    If  each  child  receives  6  cents,  48 
cents  could  be  divided  among  as  many  children  as 
the  times  that  6  cents  could  be  taken  from  48  cents, 
namely,  8  times  :  hence,  among  8  children. 

Both  forms  of  division  must  be  made  clear  to  the 
pupils  through  practical  problems ;  for  both  forms 
are  of  equal  use. 

Division,  as  soon  as  it  deals  with  numbers  beyond 
the  multiplication  table,  is  a  very  complicated  pro- 


DIVIDING   BY   TWO.  IOI 

cess ;  hence  it  is  necessary  to  be  very  patient  in 
teaching  it,  and  to  proceed  very  gradually  from  the 
easier  to  the  more  difficult.  If  the  first  difficulties 
are  really  overcome,  much  has  been  done  to  lighten 
the  subsequent  work. 

26.   DIVIDING  BY  Two. 

FIRST    EXERCISE. 

Let  the  children  add  2  successively  to  2,  4,  etc.,  so 
as  to  form  the  numbers  2,  4,  6,  8,  lo,  etc.,  to  20. 

Question  thus  ?  How  many  are  2X2?  3X2?  etc. 
Two  in  2  how  many  times  ?  In  4  ?  In  6  ?  etc.  How 
many  times  can  2  be  taken  from  2  ?  From  4  ?  etc. 
How  many  twos  in  2  ?  In  4  ?  etc. 

Give  this  question  :  Two  children  are  to  divide  12 
cents  equally  ;  how  many  will  each  child  receive  ? 

Although  the  children  are  prepared,  from  what 
they  have  already  learned,  to  answer  this  and  similar 
questions,  yet,  partly  to  prepare  them  for  the  suc- 
ceeding stage,  and  partly  to  show  the  teacher  by  an 
example  how  to  manage  when  the  difficulties  involved 
appear  in  a  new  place,  we  will  explain  the  process  of 
working.  In  this  example  the  teacher  may  use  the 
cents  themselves  first,  then  marks  upon  the  board. 
The  latter  may  be  arranged  as  those  below.  Having 
written  A  and  B,  place  first  a  circle  for  a  cent  which 
A  is  to  take,  then  under  it  one  for  a  cent  which  B 
takes,  and  so  on  till  the  12  are  represented. 


102  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

A.  o  o  o  o  o  o 

B.  o  o  o  o  o  o 

Each  has  taken  6  cents.  When  a  number  is 
divided  into  2  equal  parts,  each  part  is  a  half.  The 
half  of  12  cents  is  6  cents ;  the  half  of  12  is  6. 

In  the  same  way  develop  the  idea  of  the  half  of  2, 
4,  6,  8,  10,  12,  14,  1 6,  1 8,  20. 

SECOND    EXERCISE. 

Two  children  have  15  apples,  how  many  has  each? 
Of  14  apples  each  has  7    apples  ; 
"      i  apple      "       "      \  apple. 

"    15  apples    "       "     7|  apples. 

In  the  same  way  treat  3,  5,  7,  9,  n,  13,  15,  17, 
and  19. 

THIRD    EXERCISE. 

Draw  on  the  board  two  rows  of  circles  with  10 
circles  in  each  row.  This  will  show  that  half  of  2 
rows  is  i  row ;  half  of  2  tens  is  I  ten ;  half  of  20 
is  10. 

In  the  same  way  may  the  idea  of  half  of  20,  40,  60, 
80,  and  100  be  developed. 

The  numbers  2,  4,  6,  8,  10,  12,  14,  16,  18,  20,  40, 
60,  80,  and  100  can  be  divided  immediately,  that  is, 
without  being  separated  into  parts,  because  they 
appear  in  the  twos  of  the  multiplication  table,  if  we 
regard  20,  40,  etc.,  as  2  tens,  4  tens,  etc.  Numbers 
which  do  not  so  appear  must  be  separated. 


DIVIDING   BY   TWO.  IOJ 

FOURTH    EXERCISE. 

Two  persons  together  have  4  ten-cent  pieces  and 
8  cents  ;  how  shall  they  divide  them  ? 

Each  person  takes  2  dimes  and  4  cents,  equal  to 
24  cents  ;  so  half  of  48  is  24. 

Or  the  teacher  may  write  on  the  board  4  rows  of 
10  circles  each,  and  8  circles.  Half  of  4  rows  is  2 
rows ;  half  of  8  circles  is  4  circles ;  half  of  4  tens  is 
2  tens ;  half  of  40  is  20 ;  half  of  8  is  4 ;  therefore, 
half  of  48  is  24. 

So  may  be  developed  the  idea  of  half  of  those 
numbers  whose  tens  and  units  are  even  numbers  — 
22,  24,  26,  28 ;  42,  44,  46,  48 ;  62,  64,  66,  68 ;  82,  84, 
86,  88. 

FIFTH    EXERCISE. 

Two  persons  have  3  dimes ;  how  can  they  be 
divided  ?  Each  takes  I  dime,  equal  to  10  cents. 
They  then  exchange  the  other  dime  for  10  cents,  and 
each  takes  5  cents ;  so  that  each  has  1 5  cents. 

Or  the  teacher  may  draw  on  the  board  3  rows  of 
10  circles  each.  Half  of  2  rows,  or  20,  is  10 ;  and 
half  of  the  other  row  is  5  ;  so  that  the  half  of  30  is  15. 

Treat  50,  70,  and  90  in  the  same  way. 

SIXTH    EXERCISE. 

Show  on  the  numeral  frame  3  rows  of   10  balls 
each,  and  i  row  of  6  balls.     What  is  half  of  them  ? 
Half  of  2  tens  is    I  ten,  and  the  other  ten  balls 


104  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

added  to  the  6  make  16  balls.  Half  of  1  6  is  8  ;  so 
half  of  36  is  i  ten  and  8,  or  18. 

Or  this  :  Divide  3  dimes  and  6  cents  equally  be- 
tween two  persons.  Let  each  take  I  dime  ;  exchange 
the  other  dime  for  10  cents,  which  added  to  6  cents 
make  16  cents.  Let  each  take  8  cents,  which  with 
the  dime  make  18  cents. 

So  treat  32,  34,  36,  38  ;  52,  54,  etc.  ;  72,  74,  etc.  ; 
92,  94,  etc. 

SEVENTH    EXERCISE. 

What  is  half  of  49  apples  ? 

Half  of  40  apples  is  20  apples  ; 
8       "  4  apples; 

i       "  i  apple. 

"         49  apples  is  24^-  apples. 

So  treat  all  numbers  which  have  even  tens  and 
odd  units:  21,  23,  25,  27,  29;  41,  43,  45,  47,  49;  61, 
63,  etc.  ;  8  1,  83,  etc. 

EIGHTH    EXERCISE. 

What  is  half  of  57? 

Half  of  40  is    20  ; 
16  "     8; 

i   " 


57 

Treat  in  the  same  way  all  numbers  whose  tens  and 
units  are  odd  numbers:  33,  35,  37,  39;  53,  55,  etc.; 
73>  75>  etc.  ;  93,  95,  etc. 


DIVIDING  BY   TWO.  10$ 

ILLUSTRATIVE    EXAMPLES. 

We  will  show  by  an  example  of  the  last  exercise 
(exercise  eight)  what  the  full  treatment  of  a  problem 
in  division  should  be,  as  it  has  been  developed  in  the 
successive  stages  of  work  in  division.  Division  re- 
quires a  series  of  conclusions,  and  in  this  fact  lies 
the  difficulty  which  it  presents  to  the  children.  There 
is  no  cause  for  discouragement,  however ;  for  if  divis- 
ion by  2  is  thoroughly  mastered,  the  remaining 
numbers  can  be  passed  over  much  more  rapidly.  Do 
not  introduce  the  children  to  the  formal,  written 
representation  of  the  process  till  they  have  attained 
considerable  facility  in  explaining  it.  If  they  need 
to  be  occupied  with  written  work,  there  is  material 
enough  in  the  review  of  what  precedes,  —  especially 
in  addition,  subtraction,  and  multiplication.  Proba- 
bly it  will  take  from  four  to  six  weeks  to  ground  a 
class  thoroughly  in  division  of  numbers  below  100 
by  2.  Division  by  the  numbers  from  3  to  10  will 
scarcely  require  more  time.  The  successive  steps  in 
the  solution  of  a  question  in  the  division  of  a  number 
by  2  when  both  the  tens  and  units  are  odd  numbers 
may  be  brought  out  thus  : 

(1)  Teacher.    We  will  find  the  half  of  75.     Can  we 
divide  75  immediately ;  that  is,  all  at  once  ? 

Scholar.    We  cannot  divide  75  immediately. 

(2)  T.    Why  not? 

S.  Because  75  is  not  found  in  the  twos  of  the 
multiplication  table. 


106  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

(3)  T.    Can  we  divide  7  tens  immediately  ? 
5.    We  cannot  divide  7  tens  immediately. 

(4)  T.    Why  not  ? 

vS.    Because    7  is   not   found  in  the    twos   of   the 
multiplication  table. 

(5)  T.    What  is  the  next  number  below  7  that  is 
found  there  ? 

5.    Six  is  the  next  number. 

(6)  T.    What  is  the  half  of  6  tens,  or  60  ? 
5.    Half  of  6  tens,  or  60,  is  3  tens,  or  30. 

(7)  7!   How  many  of  75  remain  to  be  divided  when 
we  have  divided  60  ? 

5.    Fifteen  remain  to  be  divided. 

(8)  T.    Can  we  divide  15  immediately? 
S.    We  cannot  divide  1 5  immediately. 

(9)  T.    Why  not? 

vS.    Because  15  is  not  found  in  the  twos  of   the 
multiplication  table. 

(10)  T.    What  is  the  next  number  below  15  that  is 
found  there  ? 

S.    Fourteen  is  the  next  number. 

(11)  T.    What  is  half  of  14? 
5.    Half  of  14  is  7. 

(12)  T.    How  many  still  remain  to  be  divided? 
5.    One  still  remains  to  be  divided. 

(13)  T.    What  is  half  of  i? 
5.    One-half  is  half  of  i. 

(14)  T.    How  many  did  we  at  first  obtain,  when 
we  divided  60  ? 


DIVIDING  BY   TWO.  IO/ 


5.    We  at  first  obtained  30. 

(15)   T.    Then  how  many  ? 

5.    Then  7. 

(i  6)    T.    Then? 

5.    One-half. 

(17)   T.    Add  them  all  together. 


(18)   T.    Give  the  entire  solution. 
5.    What  is  half  of  75  ? 

Half  of  60  =  30. 
Half  of  14=  7. 
Half  of  i  - 


Half  of  75  - 

This  example  will  be  sufficient  to  show  how  com- 
plicated is  the  process  of  division,  and  how  very 
nicely  the  work  should  be  graded,  so  as  to  lead  the 
pupils  to  ask  and  answer  by  themselves  all  the  neces- 
sary questions.  At  first  the  teacher  asks  the  ques- 
tions ;  but  soon  the  brighter  pupils  may  act  as  teach- 
ers. They  may  take  their  places  in  turn  before  the 
class,  and  question  their  fellow  pupils.  This  must  be 
continued  till  all,  even  the  dull  ones,  are  able  to  do 
the  same.  The  brightest  children  may  be  set  to 
questioning  single  rows  or  small  divisions.  Gradu- 
ally the  pupils  will  begin  to  unite  the  several  succes- 
sive processes  ;  at  first  two,  and  finally  all.  It  is  the 
same  here  as  in  other  complex  mental  processes :  at 
first  we  perform  the  successive  steps  consciously ; 


108  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

and  then,  as  they  are  repeated,  we  seem  to  omit 
more  or  less  of  the  intervening  steps  and  to  reach 
the  conclusion  at  once.  To  make  this  result  possible, 
however,  it  is  absolutely  necessary  that  each  step 
be  not  only  expressed  but  understood.  Hence  the 
importance  of  thorough  illustration  and  also  of  well- 
graded  and  abundant  practice. 

When  the  pupil  has  been  through  all  the  work  in 
division  of  numbers  by  2  which  has  now  been  pointed 
out,  so  that  he  is  prepared  independently  to  arrange 
the  conclusions  in  order,  he  may  be  put  at  written 
work.  The  form  of  this  may  be  the  following  : 

\  of  75.  2  in  75. 

\  of  60  =  30.  2  in  60  =  30  times. 

\  of  14  =    7.  2  in  14  =    7  times. 

\  of    i  =      \.  2  in     i  =      \  times. 


of  75  =  371  2  in  75  =  y\  times. 


Such  work  as  this  is  the  best  preparation  for  the 
division  of  larger  numbers  ;  but  then  the  written 
form  must  be  the  result  of  a  thorough  comprehen- 
sion. It  is  of  importance  that  every  figure  stands  in 
its  proper  place.  For  this  purpose  it  is  well  to  divide 
the  slates  into  little  rectangles,  and  to  have  one 
figure  put  in  each.  The  written  solution  of  a  prob- 
lem should  be  a  picture  of  order.  The  orderly 
arrangement  of  the  work  makes  it  possible  to  dis- 
pense with  many  words  ;  it  tends  to  mathematical 
brevity  and  definiteness  ;  and  it  materially  shortens 
the  teacher's  work  of  correction. 


DIVIDING  BY   TWO.  1  09 

If  the  division  of  small  mumbers  is  to  be  a  prepa- 
ration for  the  division  of  larger  numbers,  the  forms 
which  represent  the  two  processes  should  agree.  The 
arrangement  of  the  written  work,  in  case  of  large 
numbers,  is  the  following  : 

2)75(37i 
6 

IS 


The  separation  of  the  dividend  into  parts,  in  the 
form  given  above,  is  sufficiently  like  this  to  give  no 
trouble  in  later  work  ;  but  any  other  separation  —  as 
into  70,  4,  and  i,  or  into  72,  2,  and  I,  because  the 
children  happen  to  know  the  half  of  these  numbers  — 
would  be  wrong  practice,  because  it  wottld  not  be  a 
preparation  for  higher  work  ;  for  in  the  division  of 
large  numbers  the  result  is  to  be  reached  figure  "by 
figure,  and  hence  the  same  should  be  true  in  the 
division  of  small  numbers.  The  experienced  arith- 
metician, especially  when  his  work  is  wholly  mental, 
is  bound  by  no  rules.  He  often  reaches  the  result 
by  a  short  cut.  But  it  is  never  to  be  lost  sight  of 
that  it  is  the  business  of  this  stage  of  the  work  to 
develop  the  power  of  arithmetical  calculation.  It 
will  be  time  enough  later  to  teach  shorthand  pro- 
cesses. 

Then,  too,  the  importance  of  performing  the  work 


I  10          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

at  one  stage  in  such  a  way  as  to  prepare  the  pupils 
for  subsequent  stages  is  too  often  overlooked.  It  is 
much  easier  to  make  children  comprehend  the  pro- 
cesses of  division,  as  they  stand  related  to  one  an- 
other, when  dealing  with  numbers  completely  within 
their  comprehension  than  in  the  treatment  of  incom- 
prehensible numbers ;  hence  the  importance  of  slow 
and  well-graded  progress  in  division  of  small  numbers 
by  2. 

The  young  teacher  should  not  be  impatient  if 
much  time  is  spent  with  the  number  2  ;  for  if  the 
fundamental  conceptions  of  division  are  here  made 
clear,  subsequent  progress  will  not  only  be  more 
rapid,  but  much  freer  from  that  confusion  which 
results  from  an  attempt  to  teach  a  new  principle  in 
connection  with  indistinctly  formed  ideas. 

27.   DIVIDING  BY  THREE. 

First  develop  through  illustrations  —  as,  for  exam- 
ple, balls  on  the  numeral  frame,  sticks,  buttons,  or 
marks  of  various  kinds  on  the  board  —  the  third  part 
of  3,  6,  9,  12,  15,  18,  21,  24,  27,  30,  60,  90;  then  the 
third  part  of  i  and  2.  The  latter  may  be  done  by 
dividing  a  piece  of  paper  into  3  equal  parts,  and 
naming  each  part  \ ;  then  by  treating  a  second  piece 
in  the  same  way.  A  practical  problem  may  be  used 
for  the  same  purpose :  three  children  have  2  apples 
to  divide  among  them  ;  what  part  does  each  receive  ? 


DIVIDING  BY   THREE.  Ill 

They  cut  one  apple  into  3  equal  parts,  and  each 
takes  i  part,  or  \  of  an  apple.  They  then  treat  the 
second  in  the  same  way.  Each  child  then  has  f  of 
an  apple ;  so  \  of  2  apples  is  |  of  i  apple.  Or,  lay 
two  equal  circles  of  paper  one  upon  the  other,  and 
cut  them  into  3  equal  parts.  Each  double  part  is  \ 
of  the  2  pieces,  or  f  of  i  piece. 

In  the  development  of  the  process  of  dividing  by  3 
it  will  not  be  necessary  to  divide  the  work  into  stages 
so  carefully  graded  as  in  the  treatment  of  division 
by  2.  I  will,  however,  indicate  the  corresponding 
stages,  as  they  are  always  useful  in  the  treatment  of 
dull  pupils.  They  are  the  following  : 

a.  3,  6,  9,  12,  15,  1 8,  21,  24,  27,  30,  60,  90. 

b.  i,  4,  7,  10,  13,  1 6,  19,  22,  25,  28. 

c.  2,  5,  8,  n,  14,  17,  20,  23,  26,  29. 
d-  33>  36,  39>  63>  66,  69,  93,  96,  99. 

e.  42,  45.  48,  Si.  54,  57>  72,  75,  78,  81,  84,  87. 

f.  The  rest  of  the  numbers  below  100. 

An  example  will  show  the  agreement  of  the  treat- 
ment with  that  of  dividing  by  2. 

Three  boys  are  to  divide  89  cents  among  them  ; 
they  have  8  dimes  and  9  cents.  How  many  cents 
does  each  receive  ? 

Teacher.    How  many  dimes  can  each  take  ? 

Scholar.     Each  can  take  2  dimes. 

T.    What  will  be  left  ? 

5.    2  dimes  and  9  cents  =  29  cents. 


112  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

T.  Can  they  divide  29  cents  at  once  ? 

5.  No,  for  29  is  not  found  in  the  threes  of  the 
multiplication  table. 

T.  What  is  the  next  number  below  29  that  is 
found  there  ? 

*£.  27  is  the  next  number. 

T.  What  is  a  third  of  27  ? 

5.  9  is  a  third  of  27. 

T.  How  many  cents  has  each  now  ? 

5.  20  +  9  =  29. 

T.  How  many  cents  are  still  to  be  divided  ? 

5.  2  cents. 

T.  How  much  does  each  receive  of  2  cents  ? 

5.  |  of  a  cent. 

T.  How  many  cents  has  each  altogether? 

5.  20  +     + 


The  pupil  must  finally  be  brought  to  the  point 
where  he  can  give  the  following  solution  in  sub- 
stance alone  : 

We  cannot  divide  89  immediately  by  3,  because  it 
is  not  found  in  the  threes  of  the  multiplication  table. 
The  next  number  found  there  below  89  is  60.  A 
third  of  60  =  20.  Now  29  remains  to  be  divided. 
As  29  is  not  in  the  threes  of  the  multiplication  table, 
we  divide  the  next  lower  number,  27.  A  third  of 
27  is  9.  We  have  now  divided  87,  and  2  remains  to 
be  divided.  A  third  of  2  =  f.  So  that  the  third 
part  of  89  =  20  -I-  9  +  f  =  29^. 


DIVIDING  BY  OTHER  NUMBERS.  113 

Written  exercises  in  dividing  by  3  are  not  to  be 
assigned  to  the  pupils  till  they  have  gained  a  thorough 
understanding  of  the  matter  and  some  facility  in 
solving  problems.  The  written  expression  for  the 
example  just  given  will  appear  as  follows  : 

i  of  89.  3  in  89. 

\  of  60  =  20.  3  in  60  =  20  times. 

\  of  27  ^:    9.  or,          3  in  27  =    9  times. 

\  of    2  -     f .  3  in    2  =    f  times. 

-J-  of  89  =  2gf.  3  in  89  =  29!  times. 

Which  form   is    to   be  used  will   depend   upon    the 
special  statement  of  the  question  and  its  solution. 


28.   DIVIDING  BY  OTHER  NUMBERS  TO  10. 

After  the  treatment  of  division  by  2  and  3  it  will 
not  be  necessary  to  go  into  particulars  in  regard  to 
dividing  by  4,  5,  6,  7,  8,  9,  and  10.  The  same  method 
is  to  be  followed  with  all  these  numbers.  When 
they  have  all  been  taught,  the  pupils  may  be  re- 
quired, by  way  of  review,  to  divide  a  number  by  each 
of  the  fundamental  numbers  in  turn.  Great  facility 
should  be  attained  in  the  division  of  those  numbers 
which  are  of  special  importance  in  business  ;  as,  12, 
24,  25,  30,  50,  60,  100. 

It  will  be  observed  that  division,  as  indicated 
above,  depends  upon  the  separation  of  numbers  into 

tens  and  units,  although   the  words   tens  and  units 

' 

+^         OF  THE 


.. 


114  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

have  been  for  the  most  part  avoided.    .Let  us  divide 
89  by  three,  using  these  terms  : 

89  consists  of  8  tens  and  9  units.  A  third  of  8 
tens  is  2  tens,  with  a  remainder  of  2  tens,  which  are 
equal  to  20  units.  Add  9  units  and  the  sum  is  29 
units.  A  third  of  29  units  is  9  units,  with  a  remain- 
der of  2  units.  A  third  part  of  2  units  is  |.  Hence, 
\  of  89  =  2  tens  +  9  units  +  f  r=  29!. 

This  solution  requires  more  statements  than  the 
one  given  above ;  and  there  is  danger  that  some  of 
these  may  escape  the  memory.  Frequently  the  pro- 
cess is  shorter  if  the  number  to  be  divided  is  sepa- 
rated into  the  parts  indicated  in  the  first  solution ; 
as,  89  =  60  +  27  +  2.  The  separating  of  numbers  in 
this  way  is  the  most  important  part  of  division. 

29.    PRACTICE  WORK. 

In  the  preceding  work  on  division  it  must  appear 
that  there  is  a  necessity  for  the  most  careful  separa- 
tion of  the  work  into  stages  founded  upon  the  degree 
of  difficulties  to  be  overcome ;  so  that  the  work  will 
conform  to  the  educational  maxim  :  From  the  easier 
to  the  more  difficult.  First  come  the  numbers  found 
within  the  multiplication  table  ;  then  follow  those 
without  this  table,  but  divisible  without  a  remainder ; 
and  finally,  those  numbers  which  lie  beyond  the 
table,  but  are  not  divisible  without  a  remainder. 

Some  of  the  work  in  the  division  of  numbers  below 


PRACTICE    WORK.  115 

a  hundred  is  no  doubt  more  difficult  than  work  with 
numbers  between  100  and  1,000;  and  yet  these  are 
fundamental  difficulties,  and  it  is  better  to  conquer 
them  in  connection  with  small  numbers. 

In  some  cases,  however,  it  may  be  well  to  limit  the 
division  of  numbers  at  this  stage,  and  not  apply  it  to 
all  numbers.  The  following  is  a  good  selection  of 
numbers  for  this  purpose  : 

By    2  divide  numbers  from  i  to    20. 

(t  3  ((  (t  ((  J  ((  OQ 

«  ^          tt  (t  t<          j      U          ^o 

«  H  ((  ((  (t  -r         (i  gft 

"     6     "  "  "     i   "     60. 

((  tj  It  ((  <(  T  ft  <TQ 

"     8     "  "  "     i   "     80. 

"     9     "  "  "     i  "     90. 

"    10     "  "  "     i  "  100. 

Chart  XII.  will  be  found  useful  at  this  stage, 
because  it  contains  all  numbers  below  100,  so  arranged 
that  they  can  be  readily  assigned  for  practice  in 

division :      „  .  , 

By    2,  series  a,  b. 

"  3,  "  *,  *,  c. 

"  4,  "  a,  b,  c,  d. 

"  5,  "  a,  b,  c,  d,  e. 

"  6,  "  a,b,c,dye,f. 

"  7,  "  a,b,c,d,e>f,g. 

«  8,  «  a,b,c,d,e,f,g,h. 

"  9,  "  a,  b,  c,  d,  e,  /,  g,  h,  /. 

"  10,  "  ^,  ^,  c,  d,  e,f,  g,  k,  /,  k. 


Il6          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

The  number  100  should  receive  special  considera- 
tion at  this  time.  One  kind  of  exercises  is  the  rind- 
ing of  every  two  parts  of  which  100  consists ;  for 
example : 

99  +    i,      98  +  2,      97+    3,     96  +  4,       95+    5, 

94+   6,       93  +  7>       92+    8>     9J+9>       9°+IO> 
89  +  1 1,  and  so  on  to    50  +  50. 

Another  kind  of  exercises  is  the  following  :  100  is 
equal  to 

i  x  100,       2  x  50,  3  x  33  +  i,      4  x  25, 

5X    20,       6x16  +  4.     7x14  +  2,      8x12  +  4, 
and  so  on  to  50  X    2. 

A  third  kind  of  work,  which  ought  to  be  done 
before  passing  on  to  the  treatment  of  numbers  to 
1,000,  is  making  change  from  a  dollar  for  any  smaller 
sum ;  for  example :  33  cents  from  a  dollar ;  the 
change  may  be  2  cents,  a  nickel,  a  dime,  and  half  a 
dollar,  or  2  quarters. 


NUMBERS  FROM   ONE    TO  A    THOUSAND.      II 7 

CHAPTER   IV. 

NUMBERS   FROM  ONE   TO   A   THOUSAND. 

3O.    COUNTING  AND  WRITING. 

101  to  200. 

FOR  the  same  reason  that  it  was  thought  best  for 
pupils  to  be  made  acquainted  with  numbers  from 
eleven  to  twenty  before  studying  numbers  from 
twenty  to  one  hundred,  it  is  here  recommended 
that  they  be  made  somewhat  familiar  with  numbers 
from  101  to  200  before  they  are  required  to  deal  at 
all  with  numbers  from  201  to  1,000.  The  following 
steps  will  bring  the  pupils  to  the  desired  result : 

1.  Counting  from  101  to  200. 

2.  Counting  from  200  to  101. 

3.  Writing  numbers  from  101  to  200. 

4.  Reading  numbers  written  on  the  board  by  the 

teacher. 

5.  Writing  the  numbers  from  dictation. 

6.  Separating  the  numbers  from  101  to  200,  into 

a.    Hundreds,  tens,  and  units,  as,  for  example : 

134—1  hundreds,  3  tens,  and  4  units. 
150-=!         "          5     "        "    o      " 
105  =  1         "          o     "       "    5      " 
200  =  2        "         o    "       "    o     " 


Il8  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

b.    Tens  and  units,  as,  for  example  : 

134=  13  tens,  and  4  units. 

150=15     "        "    o      " 

105  =  10     "        "    5      " 

200  =  20     "        "    o      " 

If  the  work  on  numbers  below  one  hundred  has 
been  thoroughly  done,  there  will  be  need  of  but  little 
objective  teaching  at  this  stage  of  the  work.  It 
would  be  well  for  the  teacher  to  be  provided  with  a 
few  strings  of  one  hundred  buttons  each,  or  a  few 
bunches  of  sticks,  each  bunch  containing  one  hun- 
dred;  so  that  he  can  illustrate  his  work  objectively, 
and  afford  an  opportunity  for  the  dull  pupils  to 
handle  the  objects  themselves.  But  it  is  by  no 
means  necessary  to  present  all  the  numbers  at  this 
stage  of  the  work  in  the  form  of  objects  in  the  hands 
of  every  child.  Too  much  objective  teaching  of  num- 
bers is  only  less  stultifying  than  too  little.  It  is  now 
time  to  appeal  to  the  imagination  and  to  the  power  of 

abstraction. 

201  to  1,000. 

The  pupils  should  now  be  taught  to  count  by  hun- 
dreds to  1,000  ;  then  the  numbers  201,  etc.,  to  1,000 
should  be  treated  in  the  same  way,  and  by  the  same 
steps  as  were  recommended  in  the  case  of  numbers 
from  101  to  200,  including  the  separation  of  the 
numbers  into  hundreds,  tens,  and  units,  as  shown 
above  under  a  and  b. 

This  work  of  counting,  reading,  writing,  etc.,  should 


ADDITION.  119 

be  continued  till  the  pupils  have  clear  ideas  of  all 
numbers  below  1,000,  know  them  as  composed  of 
units,  tens,  and  hundreds,  and  know  how  their  com- 
ponent parts  are  represented  by  figures.  The  test 
of  this  last  item  of  knowledge  is  the  ability  of  the 
pupil  to  select  the  groups  of  objects,  and  the  single 
objects,  for  which  the  different  figures  of  any  number 
below  1,000  stand.  When  this  test  can  be  easily 
borne,  it  is  time  to  advance  to  the  different  funda- 
mental operations,  but  not  before. 

31.   ADDITION. 

The  more  thoroughly  pupils  are  drilled  in  adding 
numbers  expressed  by  two  figures,  that  is,  numbers 
below  a  hundred,  the  easier  will  be  the  work  of  learn- 
ing the  addition  of  numbers  represented  by  three 
figures.  Hence  it  is  well  at  this  point  to  make  a 
thorough  review  of  the  addition  of  numbers  below  a 
hundred.  For  this  purpose  Chart  XII.  will  be  found 
very  convenient.  If  all  numbers  from  eleven  to  one 
hundred  are  in  turn  added  to  each  of  the  numbers 
on  the  chart,  the  pupils  will  have  90  times  100,  or 
9,000  examples  ;  while  the  teacher  will  be  spared  the 
labor  of  copying  any  of  them  on  the  board.  The  fol- 
lowing suggestions  are  offered  as  to  the  proper  stages 
of  the  review  here  recommended. 

Let  the  pupils  add  the  numbers  written  below  to 
each  of  the  numbers  on  Chart  XII.,  proceeding  from 
left  to  right : 


120          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

First  Stage,     a  10,     b  20,     <:  30,     d  40  -    -  100. 


Second   " 

ii, 

21, 

3*i 

41  - 

-    91- 

Third      " 

12, 

22, 

32, 

42  - 

-    92. 

Fourth    " 

13. 

23, 

33, 

43  - 

-    93- 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

Those  exercises  should,  as  far  as  possible,  be  per- 
formed orally ;  but,  when  necessary,  the  pupils  may 
be  required  to  indicate  the  work  on  their  slates.  Sup- 
pose it  is  required  to  add  the  number  47  to  each  of 
the  numbers  on  Chart  XII.,  the  work  will  appear  on 
the  slate  in  the  following  order : 

1+47-   48  76  +  47-123  43+47-   90 

12+47-    59  84  +  47-131  52  +  47-   99 

26  +  47-   73  96  +  47-143  70  +  47-117 

34  +  47-    81  2  +  47-   49  73  +  47  =  120 

47  +  47-   94  19  +  47=   66  87  +  47-134 

53+47-100  22  +  47-   69  93+47-140 

67+47  —  114  31+47—    78  etc.    etc.    etc. 

Or  the  pupils  may  be  required  to  write  the  results 
only ;  then  the  results  of  the  above  additions  would 
assume  this  form  : 

a.   b.   c.   d.   e.   f.   g.    h.    i.   k. 
I.  48  59  73  81  94  100  114  123  131  143 
m.  49  66  69  78  90   99  117  120  '134  140 
n.  etc.,  etc. 

It  will  be  noticed  that  the  numbers  to  be  added 
first  are  those  consisting  of  tens  only,  then  those 
consisting  of  tens  and  units.  In  adding  the  latter 


ADDITION.  121 

class  of  numbers,  the  tens  should  be  added  first  and 
then  the  units  ;  for  example  :  95  +  47  ;  95  +  40  =  1 35  ; 
I35  +  7  =  I42;  hence,  95+47-142. 

The  pupils  should  acquire  considerable  facility  in 
adding  numbers  of  two  places  before  proceeding  to 
the  addition  of  larger  numbers.  When  they  are  pre- 
pared to  advance  to  work  with  larger  numbers,  they 
should  be  assigned  examples  in  the  following  order : 

a.  Both  numbers  containing  tens  only ;  as, 

40  +  30-    70. 
80  +  70—1 50. 

b.  One   number   containing   tens   and   units,   the 

other  tens  only  ;  as, 

45  +  30-    75. 

48  +  60-  1 08. 

c.  Both  numbers  containing  tens  and  units  ;  as, 

43  +  24-   67. 
86  +  75  —  161. 

d.  One   number   containing   hundreds,    tens,  and 
units,  and  the  other  only  tens  and  units ;  as, 

238  +  46-284. 
475  +  48  =  523. 

e.  Both  numbers  containing  hundreds,  tens,  and 

units ;  as, 

425+248-673. 

436  +  398  ===  834- 
The  above  is  the   order  in  adding  two  numbers 


122  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

only ;  the  addition  of  columns  of  numbers  comes 
later.  This  work  is  all  to  be  done  mentally  before 
any  part  of  the  result  is  written.  In  'adding  two 
numbers,  the  following  rule  is  universally  to  be  fol- 
lowed : 

The  first  number  is  not  to  be  separated  into  parts ; 
when  the  second  consists  of  tens  and  units,  the  tens 
are  to  be  added  first,  then  the  units ;  when  it  con- 
sists of  hundreds,  tens,  and  units,  the  hundreds  are 
to  be  added  first,  then  the  tens,  and  lastly  the  units, 
e.g. : 

367  +  86;  367  +  80-447;  447  +  6-453.  378  + 
285;  378  +  200  =  578;  578  +  80  =  658;  658+5- 
663. 

Only  a  few  examples  like  the  last  should  be  given ;; 
and  these  mainly  to  the  ablest  pupils. 

The  pupils  should  have  enough  work  like  the 
above,  on  pure  numbers,  to  make  them  familiar 
with  the  processes  of  addition  ;  and  then  the  knowl- 
edge and  power  thus  gained  should  be  applied  to  the 
solution  of  simple  practical  problems.  Indeed,  some 
practical  problems  should  be  given  with  almost  every 
lesson. 

32.   SUBTRACTION. 

Before  beginning  the  subtraction  of  numbers 
above  a  hundred,  the  subtraction  of  numbers  below 
a  hundred  should  be  thoroughly  reviewed.  When. 


SUBTRACTION. 

this  has  been  done,  a  judicious  use  of  Chart  XII. 
will  save  the  teacher  much  time  and  labor. 

In  the  first  stage  of  this  work  the  minuend  should 
not  exceed  200.  If,  now,  the  pupil  is  taught  to  think 
of  each  number  on  Chart  XII.  as  100  larger  than  it 
is,  and  to  use  these  increased  numbers  as  minuends, 
each  number  on  the  Chart  may  be  used  as  a  subtra- 
hend, and  thus  the  teacher  will  have,  ready  made, 
100  x  100,  or  10,000,  examples  in  subtraction. 

These  examples  should  be  assigned  in  the  following 
order  : 

a.  The  subtrahend  containing  tens  only  ;  as, 

101  —  20  101  —  60 

112  — 20  112  —  60 

etc.,  etc.  etc.,  etc. 

b.  The  subtrahend  containing  tens  and  units  ;  as, 

101—24  101—67 

112  —  24  112  —  67 

etc.,  etc.  etc.,  etc. 

Any  number  from  201  to  1,000  may  be  used  as  a 
minuend,  and  each  of  the  numbers  on  Chart  XII.  as 
a  subtrahend,  and  thus  we  have  80,000  more  exam- 
ples in  subtraction  without  the  trouble  of  inventing 
them  or  writing  them  on  the  board. 

These  examples  are  all  to  be  solved  mentally. 
When  it  is  desirable  to  have  the  results  written 
down  by  the  pupils,  the  written  work  may  assume 
this  form  : 


124  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

147-      1  =  146  147-     2-145 

147  -12  =  135  147  —  19  =  I28 

147  —  26  —  121  147  —  22  =  125 
147-34-113  I47-3I-II6 
147-47-100  147-43-104 

etc.,    etc.,    etc.  etc.,    etc.,   etc. 

Or  the  work  may  be  more  briefly  represented 
thus  : 

a.    b.  c.  d.    e.  f.  g.  h.  i.  k. 

/.  146  135  121  113  100  94  80  71  63  51 

m.  145  128  125  116  104  95  77  74  60  54 
//.  etc.,  etc. 

The  rule  for  subtracting  numbers  consisting  of 
units  and  tens,  as  previously  given,  is  this  :  First 
subtract  the  tens,  then  the  units  ;  as, 

132-47;  132-40-92;  92-7-85;  so  132-47 
-85. 

The  brightest  pupils  may  be  encouraged  to  find 
new  ways  of  reaching  the  result ;  for  example  : 

47-50-3;   132-50-82;  82  +  3-85. 
47-42  +  5;   132-42-90;  90-5-85. 

If  the  work  here  indicated  on  numbers  below  200 
is  thoroughly  done,  no  special  difficulty  will  be  found 
in  the  subtraction  of  numbers  between  200  and  1,000. 
It  would  be  well,  however,  to  grade  the  work  in  the 
following  way : 


SUB  TRA  CTION.  1 2  5 

a.  The  minuend  containing  hundreds  and   tens; 
the  subtrahend  containing  tens  only ;  as, 

940  —  80  =  860 
770  —  80  —  690 
680  —  80=600 
etc.,  etc.,  etc. 

b.  The   minuend   containing  hundreds,  tens,  and 
units  ;  the  subtrahend  containing  only  tens  ;  as, 

997-60-937 
937-60-877 
877-60  =  817 
etc.,  etc.,  etc. 

c.  The   minuend    containing   hundreds,  tens,  and 
units  ;  the  subtrahend  containing  tens  and  units  ;  as, 

997-67^930 
930  —  67  —  863 
863-67-796 
etc.,  etc.,  etc. 

d.  Both  minuend  and  subtrahend  containing  hun- 
dreds, tens,  and  units  ;  as, 

993-267-726. 

In  all  cases  the  rule  for  the  subtraction  of  a  num- 
ber consisting  of  units,  tens,  and  hundreds  is  this  : 
Never  separate  the  minuend  into  parts  ;  but  subtract 
the  hundreds  of  the  subtrahend  first,  then  the  tens, 
and  last  the  units  ;  for  example  : 


126  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

993-267;   993-200  =  793;  793-60  =  733;  733 

-7-726. 

Here  the  subtraction  can  sometimes  be  most  easily 
performed  by  adding  enough  to  the  subtrahend  to- 
make  a  sum  equal  to  the  minuend ;  for  example  : 

975-388. 
388+    12—400; 
400+575-975; 
575+    12-587. 

It  is  possible  to  construct  drift  exercises  in  addition 
.and  subtraction,  by  making  the  processes  alternate, 
which  will  at  the  same  time  require  much  work  on 
the  part  of  the  pupils  and  very  little  on  the  part  of 
the  teacher.  Suppose  the  exercise  set  for  the  class 
to  be  this  :  Beginning  with  746,  alternately  add  248 
and  subtract  273.  The  work  would  assume  the  fol- 
lowing form  on  the  slates  : 

746  +  248  -  994;  994-273  =  72i. 

72 1  +  248  -  969 ;  969  -  273  -  696. 

696  +  248  -  944 ;  944  -  273  -  67 1 . 

671  +248—919;  919  —  273  —646. 

The  final  result  is  just  100  less  than  746,  the  num- 
ber with  which  the  work  began.  The  reason  is  that 
273,  the  number  to  be  subtracted,  is  25  more  than 
248,  the  number  to  be  added,  and  consequently  each 
addition  and  subtraction  diminishes  the  original  num- 
ber 25  :  and  four  additions  and  subtractions  diminish 
it  just  100. 


MULTIPLICATION.  I2/ 

The  teacher  can  readily  construct  any  amount  of 
drill  work,  such  that  the  correction  of  slates  will  be  as 
easy  as  in  the  example  given ;  for  he  has  only  to  sub- 
stitute any  other  number  for  476,  and  any  other  num- 
'bers  differing  from  each  other  by  25  for  248  and  273. 

This  kind  of  examples  is  well  adapted  to  furnishing 
every  pupil,  dull  or  bright,  with  all  the  work  he  can 
do  in  a  given  time ;  for  the  work  in  the  problem 
given  above,  is  only  to  be  continued  in  order  to  fur- 
nish 29  examples  in  addition  and  as  many  in  subtrac- 
tion, which  can  be  corrected  at  a  glance. 

While  facility  in  addition  and  subtraction  is  of  the 
greatest  importance  in  practical  business,  care  must 
be  taken  not  to  weary  and  discourage  the  pupils. 
Hence  it  is  recommended  that  problems  like  those 
given  above  be  not  introduced  too  often,  nor  contin- 
ued too  long. 

33.   MULTIPLICATION. 

Before  beginning  the  multiplication  of  numbers 
between  100  and  1,000,  the  multiplication  table 
should  be  thoroughly  reviewed.  Then  should  fol- 
low the  multiplication  of  numbers  between  10  and 
100,  which  will  be  also  partly  a  review.  The  three 
stages  of  the  work  will  be  the  following : 

a.    MULTIPLICATION   OF  TENS. 

9  X  20  =  9  x  2  tens  =  1 8  tens  =?  1 80. 
9x70  =  9x7    "     =  63     "     =630. 


128  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

The  pupils  are  to  be  familiar  with  the  changing  of 
tens  to  units.     Then  this  work  of  changing  units  to 
tens  and  of   changing  tens    to    units    may  be  soon 
omitted,  and  the  result  reached  at  once ;  as, 
9  x  20  —  1 80 ;  9  x  70  =  630. 

b.    MULTIPLICATION   OF  TENS   AND   UNITS. 
6X39  = 

6x  30-=  180 
6x    9-    54 


6x39-234 

The  above  example  illustrates  both  the  order  of 
procedure  in  purely  mental  reckoning,  and  also  a 
good  order  of  arrangement  when  the  results  of  men- 
tal operations  are  to  be  recorded. 

c.   MULTIPLICATION   OF   HUNDREDS,   TENS,  AND   UNITS. 
2X478;      2X400  =  800;      2X70=140;      800+140  = 

940 ;  2x8=16;  940  +  16  =  956. 

The  order  of  multiplication  is  first  the  hundreds, 
then  the  tens,  and  lastly  the  units.  In  multiplying 
in  the  head,  always  begin  with  the  highest  order,  and 
work  towards  the  units.  By  this  procedure  we  are 
compelled  to  repeat  the  partial  results  more  than  by 
the  reverse  process.  Then,  too,  when  results  are  to 
be  united,  they  should  be  united  as  quickly  as  possi- 
ble, so  as  to  cause  the  least  draught  upon  the  mem- 
ory. This  applies  especially  to  the  multiplication  of 
three-place  numbers. 


MULTIPLICATION.  129 

If  the  numbers  from  I  to  10  are  written  on  the 
board  in  an  irregular  order,  —  as  i,  2,  5,  8,  3,  7,  4,  9, 
6,  10, — problems  in  multiplication  can  be  easily  set 
for  a  class,  and  so  can  examples  requiring  either  addi- 
tion or  subtraction  to  be  combined  with  multiplica- 
tion. The  following  will  serve  as  suggestions  both 
for  the  invention  of  the  problems  and  for  the  arrange- 
ment of  the  work  by  the  pupils  : 


i  x  70  —  70 

1x36=   36 

2  X  70=  140 

2x36-   72 

5X70-350 

5  x  36  —  180 

8  x  70  =--  560 

8x36-288 

3  x  70  —  210 

3x36-108 

7x70-490 

7x36  —  252 

4  x  70  =  280 

4x36-144 

9  x  70  =  630 

9x36-324 

6  x  70  —  420 

6x  36  —  216 

iox  70  —  700 

10  x  36  -  360 

i  x  70  +  42  — 

i  x  36  +  27  = 

i  x  70  —  42  = 

i  x  36-27  = 

2  X  70  +  42  = 

2x36  +  27  = 

2  X  70  -  42  = 

2  X  36  -  27  = 

5  x  70  +  42  = 

5X36  +  27  = 

5x70-42  = 

5X36-27  = 

etc. 

etc. 

In  performing  the  first  of  such  exercises,  when  the 
results  are  to  be  written,  the  teacher  should  insist 
upon  having  the  work  so  arranged  as  to  show  all  the 
steps  in  the  solution  ;  as, 


130          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

5  X36+  27 
S  xso=  150 
5  x  6  =  30 


5  X27  =  180 
180  +  27  —  207 

Examples  of  the  multiplication  of  hundreds,  tens, 
and  units  will  be  but  few,  if  the  product  does  not 
exceed  1,000. 

Chart  XIII,  will  furnish  an  abundance  of  examples 
in  multiplication.  The  teacher  may  require  every 
number  on  the  chart  to  be  multiplied  by  i,  2,  etc.,  to 
10.  Whatever  tends  to  lessen  the  labor  of  the  teacher 
without  injuring  the  quality  of  his  work  is  not  to  be 
despised. 

It  is  well,  at  this  stage  of  the  work,  to  extend  the 
pupils'  knowledge  of  the  multiplication  table  to  1 1 
and  12.  For  ordinary  students  it  is  hardly  worth 
while  to  go  beyond  12. 

After  the  pupils  have  been  well  drilled  in  the  mul- 
tiplication of  pure  numbers,  special  attention  should 
be  given  to  the  solution  of  practical  problems.  It 
will  be  necessary  for  the  teacher  to  assist  the  pupils 
in  the  study  of  these  problems.  He  will  also  have 
an  excellent  opportunity  to  impart  much  practical 
information  in  regard  to  those  matters  to  which  the 
problems  refer,  and  to  drill  the  pupils  in  the  expres- 
sion and  application  of  the  numerical  ideas  which 
they  have  already  acquired,  and  of  the  practical 


MUL  TIPLICA  TION.  1 3 1 

truths  which  he  imparts.  In  the  solution  of  such 
problems  the  pupils  necessarily  receive  continuous 
drill  in  sustained  trains  of  reasoning.  The  following 
is  an  example : 

If  beans  are  sold  at  12  cents  a  quart,  or  80  cents  a 
peck,  how  much  is  saved  by  buying  a  peck  all  at 
once  rather  than  by  the  single  quart  ? 

If  one  quart  costs  12  cents,  a  peck,  or  8  quarts, 
would  cost  8  X  12  cents,  or  96  cents;  and  96  cents 
are  16  cents  more  than  80  cents  ;  therefore  16  cents 
would  be  saved  by  buying  a  peck  at  a  time. 

The  study  of  a  problem  of  this  kind  gives  the 
teacher  an  opportunity  to  impart  to  the  pupils  some 
elementary  ideas  upon  wholesale  and  retail  trade,  as 
well  as  upon  domestic  economy ;  while  the  pupil  is 
exercised  in  the  expression  of  the  relation  of  num* 
bers,  and  also  in  going  through  a  train  of  connected 
reasoning  and  its  expression. 

The  development  of  the  reasoning  power  in  con- 
nection with  the  learning  of  arithmetic  is  too  often 
undervalued.  Arithmetic  does  not  mean  simply  pro- 
ducing one  number  from  another  by  adding  and  sub- 
tracting, multiplying  and  dividing ;  but,  rather,  judg- 
ing, thinking,  reasoning.  Operations  with  numbers 
can  be  introduced  only  after  conclusions  are  reached 
through  the  power  of  thought.  A  practical  arithmeti- 
cian is  not  a  man  who  has  attained  great  skill  simply 
in  uniting  numbers  to  form  new  numbers,  skill  in 
numerical  operations ;  but  rather  one  who  knows 


132  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

how,  as  well,  to  make  the  judgments  .necessary  to 
be  used  in  the  solution  of  practical  problems.  If  a 
person  wishes  to  make  others  understand  what  he 
himself  has  clearly  thought  out,  he  must  take  pains 
to  set  it  out  in  clear  words  and  sentences.  If  instruc- 
tion in  arithmetic  is  to  result  in  something  more  than 
mechanical  skill  in  numerical  operations,  the  pupil 
must  be  practised,  at  every  stage  of  the  work,  both  in 
performing  the  steps  of  the  reasoning  processes  re- 
quired in  the  solution  of  problems  and  also  in  the 
brief,  exact,  and  definite  verbal  expression  of  such 
reasoning.  However  valuable  mechanical  skill  in 
performing  operations  upon  numbers  may  be,  it  is  of 
only  secondary  importance.  Of  much  more  impor- 
tance is  the  ability  to  perform  the  reasoning  pro- 
cesses which  lead  to  the  solution  of  practical  problems. 
This  reasoning  discovers  the  numerical  operations 
necessary  for  reaching  the  desired  result ;  and  with- 
out the  reasoning  the  operations  could  not  be  per- 
formed. 

34.    DIVISION. 

Before  beginning  the  division  of  numbers  between 
I  and  1,000,  a  careful  review  of  the  division  of  num- 
bers from  i  to  100  should  be  made ;  for  the  teacher 
cannot  too  often  remember  that  the  new  is  always  to 
be  united  with  the  old. 


DIVISION.  133 

DIVIDING   BY   TWO. 

The  first  thing  to  be  done  here  is  to  make  clear  to 
the  children,  and  then  give  them  ample  practice  in 
finding,  the  half 

a.  Of  2,  4,  6,  8,  10,  12,  14,  16,  18,  —  units. 

b.  Of  20,  40,  60,  80,  100,  1 20,  140,  1 60,  1 80,  — tens. 

c.  Of  200,  400,  600,  800,  ijOOO,  —  hundreds. 

This  done,  the  following  problem  and  solution  will 
show  the  proper  treatment  of  numbers  which  should 
follow : 

Divide  573  by  two. 

The  number  573  consists  of  5  hundreds,  7  tens, 
and  3  units.     We  must  first  divide  four  of  the  five 
hundreds ;  half  of  4  hundreds  is  2  hundreds  =  200. 
I  hundred  =  10  tens  ;    10  tens  and  7  tens  make 

17  tens.     Half  of  16  tens  is  8  tens 80 

One  ten  remains,  equal  to   10  units ;  to  which 
add  3  units,  and  we  have  13  units.     Half  of  12 

units  — 6 

Half  of  I  = i 

Half  of  573  =  286% 

In  order  to  bring  the  pupils  to  the  point  of  facility 
in  the  strictly  mental  division  of  numbers  from  I  to 
1,000,  it  is  necessary  for  them  to  use  the  greatest 
brevity  of  thought  and  expression  ;  hence  they  should 
soon  be  taught  to  separate  the  number  to  be  divided 
into  divisible  parts  without  the  use  of  the  words  "hun- 


134          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

dreds,"  "tens,"  and  "units."  The  written  expression 
of  the  work  given  above  will  then  assume  the  fol- 
lowing form  : 

*of  573  = 

\  of  400  =  200 

\  of  160=    80 

•J-of    12  =     6 


1  of  573=286^ 

From  the  preceding  we  derive  the  following  :  First 
divide  the  hundreds  which  are  divisible  without  a 
remainder;  reduce  the  hundreds  which  cannot  be 
directly  divided  to  tens,  and  to  these  add  the  tens  ; 
divide  what  of  the  tens  can  be  divided  without  a 
remainder  ;  reduce  the  rest  of  the  tens  to  units,  and 
to  these  add  the  units,  etc. 

The  following  figures  will  show  how  thoroughly 
the  foregoing  work  prepares  the  pupils  for  the  usual 
written  form  of  division  : 

573^2  = 

4-  • 

17- 
16. 

13 

12 
I 

After  the  detailed  explanation  in  regard  to  division 


DIVISION.  135 

of  numbers  from  I  to  100,  it  must  be  unnecessary  to 
go  further  into  detail  here.  If  the  practice  there 
recommended  is  here  reviewed,  the  pupils  will  now 
generally  find  no  difficulty  in  dividing  numbers  below 
1,000  by  3,  4,  5,  6,  7,  8,  9,  10,  or  by  20,  30,  40,  etc., 
to  100.  It  will  sometimes  happen,  however,  that  chil- 
dren will  come  from  an  unskilful  teacher,  or  will  be 
generally  so  dull  that  it  will  be  desirable,  at  this  stage, 
to  introduce  division  first  by  2,  then  by  3,  etc.,  to  10, 
and  to  drill  on  each  number  by  itself.  In  such  case,, 
it  is  recommended  that  the  divisible  hundreds,  tens, 
and  units  be  treated  at  first  by  themselves  ;  for  exam- 
ple, the  division,  by  3,  of 

a.  3,  6,  9,  12,  15,  18,  — units; 

b.  30,  60,  90,  120,  150,  1 80, — tens; 

c.  300,  600,  900,  —  hundreds. 


So  may  the  division  by  the  other  units  be  intro- 
duced, and  frequently  with  profit. 

Practical  problems  are  necessarily  omitted  in  these 
papers  :  but  they  are  by  no  means  to  be  omitted  from 
the  pupils'  work.  The  young  teacher  is  earnestly 
recommended  to  make  use  of  books  of  problems,  and 
not  to  rely  solely  upon  his  power  of  invention.  Such 
books  should  be  used  a  part  of  the  time  by  the  pupils 
themselves.  In  this  way  the  power  of  reading  and  of 
interpreting  the  written  page  is  developed.  A  part 
of  the  time  the  problems  should  be  read  to  the  pupils 
by  the  teacher.  If,  now,  the  teacher  adopts  the  in- 


136  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

variable  practice  of  reading  a  problem  but  once,  the 
pupils'  power  of  attention  will  be  greatly  strength- 
ened. 

The  material  for  mental  work  in  all  the  fundamen- 
tal rules  can  be  indefinitely  increased  by  the  use  of 
Chart  XIIL,  as  has  been  explained  heretofore.  The 
pupils  can  use  it  as  the  basis  of  thousands  of  exam- 
ples to  be  performed  in  school  upon  the  slate,  which 
take  the  place  of  mental  work  without  the  slate. 
More  of  such  exercises  in  the  fundamental  rules  will 
now  be  suggested. 

ADDITION. 

To  365  add  each  number  on  Chart  XIIL  That 
gives  100  problems  in  addition.  Now,  instead  of  365, 
each  number  from  101  to  1,000  may  be  used;  which 
gives  900  X  100  =  90,000  additions. 

The  correctness  of  the  results  may  be  proved  at 
the  end  of  the  hour  by  letting  the  pupils  change 
slates  and  read  the  answers  through. 

SUBTRACTION. 

Let  each  number  on  Chart  XIIL  be  subtracted 
from  365.  Then,  in  place  of  365,  use  each  number 
from  101  to  1,000,  and  we  have,  in  all,  900  X  100  = 
90,000  subtractions. 

MULTIPLICATION. 

Multiply  each  number  on  the  chart  by  2,  3,  4,  etc., 
to  12,  and  we  have  1,100  multiplications.  Write  i,  2, 


WRITTEN  AND  MENTAL   ARITHMETIC.          137 

3,  4,  etc.,  to  12,  before  each  number  on  the  chart,  and 
multiply  by  2,  3,  4,  etc.,  to  12,  and  we  have  121,000 
multiplications. 

DIVISION. 

Put  the  figure  i  before  each  number  of  the  chart, 
and  divide  the  resulting  number  by  2,  3,  4,  etc.,  to  12, 
and  there  are  1,100  divisions.  Replace  the  i  by  3,  4, 
5,  etc.,  to  12,  successively,  and  we  have  n  x  1,100  = 
12,100  divisions. 

The  ingenious  teacher  will  be  able  to  save  time  and 
labor  in  other  ways  by  the  use  of  this  chart. 

35.   WRITTEN  AND  MENTAL  ARITHMETIC. 

Heretofore  written  and  mental  arithmetic  have  not 
been  separated.  The  form  of  the  written  exercises 
has  corresponded  strictly  to  the  course  of  thought  in 
the  mental  exercises.  It  is  possible,  however,  to  man- 
age the  written  work  in  such  a  way  as  to  save  both 
space  and  time.  But,  although  this  saving  is  impor- 
tant, it  is  not  to  be  gained  at  the  expense  of  clear  un- 
derstanding. The  mind  of  the  learner  needs  to  be 
prepared  beforehand  for  obtaining  a  clear  insight  into 
the  reasons  for  the  shorter  processes  of  written  work, 
which  are  of  special  advantage  in  dealing  with  larger 
numbers  ;  and  on  this  account  their  consideration  has 
been  postponed  to  a  later  stage. 

Sometimes  the  terms  mental  arithmetic  and  written 
arithmetic  are  set  over  against  each  other,  as  though 

" 


138  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

they  stood  for  two  distinct  kinds  of  arithmetical  work. 
Such  a  division,  however,  is  incorrect ;  for  all  arith- 
metical computation  is  made  by  the  mind.  The  use 
of  the  terms  oral  arithmetic  and  memory  arithmetic 
on  the  one  hand,  and  of  slate  arithmetic  and  figuring 
or  ciphering  on  the  other,  is  often  faulty  for  the  same 
reason. 

It  is  true  that  sometimes  figures  are  used  to  assist 
the  memory  in  retaining  the  numbers  under  consid- 
eration, and  at  others  the  work  is  done  by  the  mind 
without  such  help  ;  and  perhaps  no  better  terms  have 
been  invented  to  indicate  these  two  facts  than  the  old 
names  of  written  and  mental  arithmetic.  It  is  certain 
that  the  use  of  no  other  terms  would  change  the  facts, 
or  make  the  two  processes  either  more  or  less  alike. 

What  problems  belong  to  written  arithmetic  and 
what  ones  to  mental  arithmetic  depends  upon  the 
ability  of  the  pupils  to  hold  in  the  memory  more  or 
fewer,  larger  or  smaller  numbers.  Then,  too,  a  pupil 
well  drilled  in  mental  arithmetic  will  often  solve  prob- 
lems without  the  use  of  figures,  when  others  would 
require  the  aid  of  pen  or  pencil.  There  can  no  abso- 
lute limit  to  either  class  of  problems  be  drawn.  In 
general,  however,  it  is  sufficient  if  the  problems  of 
ordinary  business,  which  do  not  involve  numbers 
larger  than  a  thousand,  can  be  solved  without  the 
aid  of  figures ;  though,  of  course,  problems  may 
sometimes  be  solved  mentally  which  involve  much 
larger  numbers. 


WRITTEN  AND  MENTAL  ARITHMETIC.        139 

It  certainly  is  well  for  all  practical  business  men  to 
be  able  to  use  readily  numbers  below  1,000,  without 
recourse  to  written  figures.  In  order  to  secure  this 
ability,  work  in  written  arithmetic,  with  its  own 
proper  methods,  has  been  deferred  to  a  later  period 
than  usual.  After  practice  in  numbers  below  1,000 
has  given  the  pupils  skill  in  mental  computation,  and 
a  clear  comprehension  of  its  principles,  insight  into 
the  principles  of  written  arithmetic  will  be  gained 
much  more  easily. 

In  order  that  the  acquired  facility  in  mental  arith- 
metic should  be  retained,  it  is  absolutely  necessary 
that  mental  arithmetic,  in  the  narrow  sense  of  the 
term,  should  be  closely  connected  with  written  arith- 
metic, whether  instruction  in  the  two  kinds  of  work 
is  given  in  the  same  or  in  different  hours. 

Written  arithmetic,  as  well  as  mental  arithmetic, 
should  not  be  practised  mechanically,  so  that  the 
operations  are  performed  merely  by  rule.  The  short- 
ened processes  of  written  arithmetic  should  be  devel- 
oped out  of  the  processes  of  mental  arithmetic  which 
have  already  been  explained.  If  this  is  done,  the 
scholar  will  come  to  know  not  only  the  processes  and 
rules,  but  their  reasons.  The  pupil  is  never  to  work 
by  a  rule,  like  a  mathematician  by  his  formula,  till  he 
understands  the  reason  for  his  procedure  and  for  the 
rule. 

War  is  to  be  continually  waged  against  all  mechan- 
ical management  of  mathematical  instruction,  and 


140          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

against  all  learning  of  facts  without  reasons  ;  yet,  in 
the  four  fundamental  operations  of  arithmetic,  the 
pupil  is  to  attain  an  ease  and  a  rapidity  of  working 
which  closely  resemble  that  of  a  perfect  machine ;  so 
that  all  his  mental  power  may  be  given  to  the  reason- 
ing processes  which  the  solution  of  the  problems 
may  require. 

If  the  work  previously  suggested  has  been  well 
done,  the  pupils  are  now  prepared  to  enter  upon  the 
stage  of  written  arithmetic  proper,  and  readily  to 
understand  its  processes. 


HIGHER  NUMBERS.  141 


CHAPTER   V. 

HIGHER  NUMBERS. 

36.    NUMERATION. 

NUMERATION  properly  means  counting ;  but  here  it 
has  an  enlarged  meaning.  It  signifies  counting,  form- 
ing higher  or  complex  units  out  of  a  definite  collec- 
tion of  less  complex,  or  simple  units,  and  also  repre- 
senting these  different  units  by  means  of  figures. 

In  studying  numbers  from  i  to  10,  i  to  20,  and  i 
to  1,000,  the  pupil  has  incidentally  learned  something 
of  the  nature  of  the  decimal  system  of  numbers,  and 
of  the  method  of  representing  numbers  by  the  Arabic 
system  of  notation ;  but  it  is  now  time  to  make  his 
knowledge  more  definite  and  systematic,  and  to  ex- 
tend it  still  farther.  For  this  purpose  review  the 
grouping  and  writing  of  numbers. 

Call  attention  to  the  fact  that  a  single  ball  on  the 
numeral  frame,  a  single  dot  on  Chart  X.,  a  single 
finger,  etc.,  is  represented  by  the  figure  i  standing 
alone ;  two  balls,  two  dots,  two  fingers,  etc.,  by  the 
figure  2  ;  three  balls,  three  dots,  three  fingers,  etc., 
by  the  figure  3  ;  and  so  on  to  nine. 

Next  show  that  a  group  of  ten  balls,  ten  dots,  etc., 
is  not  represented  by  another  figure,  but  by  the  figure 


142  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

I  standing  in  the  second  place  from  the  right ;  that 
two  groups  of  ten  balls,  ten  dots,  etc.,  each  are  repre- 
sented by  the  figure  2 ;  three  such  groups  by  the 
figure  3  ;  and  so  on  to  nine  groups.  Name  these 
groups  tens,  and  make  the  use  of  the  name  familiar 
by  counting  the  rows  of  balls  on  the  numeral  frame, 
and  the  rows  of  dots  on  Chart  X.,  etc.,  thus  :  one  ten, 
two  tens,  three  tens,  etc. 

Then  explain  that  ten  groups  of  ten  each,  or  ten 
tens,  are  called  a  hundred,  and  are  represented  by  the 
figure  i  standing  in  the  third  place ;  that  two  hun- 
dreds are  represented  by  the  figure  2  ;  three  hundreds 
by  the  figure  3  ;  and  so  on  to  nine  hundreds. 

Explain  that  ten  hundreds  are  called  a  thousand, 
and  are  represented  by  the  figure  i  placed  in  the 
fourth  place  ;  that  two  thousands  are  represented  by 
the  figure  2  ;  three  thousands  by  the  figure  3  ;  and 
so  on  to  nine  thousands. 

A  good  set  of  objects  for  illustrating  the  grouping 
of  numbers  and  the  use  of  figures  may  be  easily  made 
of  large  buttons.  Single  buttons  are  units  ;  strings 
of  ten  each,  tens ;  bundles  of  ten  strings  each,  hun- 
dreds ;  packages  of  ten  hundreds  each,  thousands. 
The  figures  may  be  written  as  the  groups  are  shown. 

Still  another  excellent  apparatus  for  this  purpose 
consists  of  ten  cubes  an  inch  on  a  side  ;  nine  sticks 
an  inch  square  and  ten  inches  long,  marked  with  lines 
an  inch  apart,  so  as  to  represent  inch  cubes  ;  and  nine 
pieces  of  board  ten  inches  square  and  an  inch  thick, 


NUMERA  TION.  1 43 

marked  off  with  lines  an  inch  apart,  so  as  to  represent 
100  inch  cubes  each.  The  cubes  are  the  units ;  the 
sticks  represent  the  tens ;  the  pieces  of  board  stand 
for  the  hundreds ;  while  all,  laid  up  in  the  form  of  a 
cube,  represent  a  thousand  small  cubes. 

Several  kinds  of  apparatus  are  better  than  any  one 
kind,  and  good  apparatus  may  be  so  used  as  to  save 
the  teacher  much  labor.  But,  somehow,  the  writing 
of  numbers  should  be  illustrated  objectively,  till  the 
pupils  can  readily  write  any  numbers  from  i  to  1,000, 
when  the  objects,  grouped  as  has  just  been  indicated, 
are  shown  them  ;  till  they  can  find  the  objects  and 
groups  of  objects  representing  any  written  number 
from  i  to  1,000;  find  the  number  of  tens  and  units 
in  any  number  of  single  objects  ;  the  number  of  units 
in  any  number  of  objects  grouped  in  tens,  as  strings 
of  buttons  ;  the  number  of  units  in  any  number  of 
tens  and  units  ;  the  number  of  hundreds  in  any  num- 
ber of  tens  ;  the  number  of  tens  in  any  number  of 
hundreds  ;  the  number  of  tens  in  any  number  of  hun- 
dreds and  tens  ;  the  number  of  hundreds,  of  tens,  and 
of  units  in  a  thousand  ;  and  the  number  of  hundreds, 
tens,  and  units,  of  tens  and  units,  and  of  tens,  in  any 
number  of  thousands,  hundreds,  tens,  and  units. 

When  all  this  can  be  readily  done  with  the  objects 
themselves,  the  pupils  should  be  drilled  in  changing 
written  numbers  into  equivalent  numbers  with  differ- 
ent groupings  ;  as,  for  example, 


144 


ARITHMETIC  IN  PRIMARY  SCHOOLS. 


1,328  —  i  thousand,  3  hundreds,  2  tens,  and  8  units. 
1,328—  13         "          2     "        "     8     " 

1,328-  132     "        "     8     " 

1,328-  1,328     " 

i 

Or  the  following : 

-4,807  —  24  thousands,  8  hundreds,  o  tens,  7  units. 
34,807-  248          "         o     "      7      " 

24,807—  2,480     "      7      " 

24,807  —  24,807      " 

It  must  be  made  perfectly  clear  to  the  pupils  that, 

a.  Units  are  represented  by  the  ist  figure. 
Tens  "  "  "     "    2d       " 
Hundreds     "             "             "     "    3d 
Thousands  "             "            "     "    4th      " 

b.  10  units          —  i  ten  ; 

10  tens  —  i  hundred  ; 

10  hundreds  —thousands,  etc. ; 

so  that  always  ten  units  of  a  lower  order  are  equal  to 
one  unit  of  the  next  higher  order. 


THOUSANDS. 

HUNDREDS. 

TENS. 

UNITS. 

i 

i 

0 

i 

0 

0 

I 

0 

0 

0 

i 

3 

2 

8 

4 

8 

0 

7 

NUMERATION.  1 45 

In  this  scheme  are  first  written   i,  10,  100,  1,000. 
Then  follows  the  explanation  that 

i  ten  —      10  units, 

i  hundred  =    10  tens  —    100  units. 

i  thousand  =  10  hundreds  =  100  tens  =  1000  units. 

Then  should  follow  the  writing  and  analysis  into 
thousands,  hundreds,  tens,  and  units  of  other  num- 
bers ;  as,  1,328,  4,807,  etc. 

The  writing  of  numbers  from  dictation,  by  the  aid 
of  this  scheme,  should  gradually  give  place  to  the 
writing  of  dictated  numbers  without  such  aid. 

If  the  writing  and  analysis  of  numbers  below  1,000 
is  thoroughly  mastered,  it  will  be  but  little  work  for 
the  teacher  to  make  clear  to  the  pupil  the  extension 
of  the  same  principles  to  numbers  above  1,000.  For 
this  purpose  the  scheme  given  above  may  be  extended 
nine  or  ten  places.  These  should  be  broken  up  into 
groups  of  three  places  each  ;  which  can  readily  be 
done  by  double  lines,  as  shown  above  between  the 
thousands  and  hundreds.  The  headings  of  the  sec- 
ond group,  or  period,  would  be,  thousands,  ten-thou- 
sands, hundred-thousands ;  and  so  of  the  millions, 
billions,  etc. 

When  this  work  has  been  well  done,  the  pupil 
needs  but  two  more  suggestions  : 

a.  To  read  any  number,  begin  at  the  right  and 
divide  it  into  periods  of  three  figures  each,  except  the 
last,  which  may  contain  three,  two,  or  one  figure  ;  read 


146  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

each  period  as  though  it  stood  alone,  adding  the  name 
of  the  last  place  in  the  period,  except  in  case  of  the 
last  period  read;  as,  24,341,101,268,  to  be  read, — 
twenty-four  billion,  three  hundred  forty-one  million, 
one  hundred  and  one  thousand,  two  hundred  sixty- 
eight. 

b.  To  write  any  number,  begin  with  the  highest 
period,  and  fill  each  subsequent  period,  using  zeros 
when  the  period  is  wholly  or  partly  omitted  ;  for 
example,  to  write  twenty-four  millions  and  seven- 
teen, put  three  zeros  in  the  thousands  period,  and  a 
zero  in  place  of  the  hundreds  in  the  units  period,  — 
24,000,017. 

The  directions  for  extending  numeration  so  as  to 
cover  all  the  higher  numbers  are  put  here  for  the 
.sake  of  completeness,  and  for  the  use  of  the  bright 
pupils  ;  but  it  is  well  to  introduce  the  writing  of 
large  numbers  gradually,  as  the  pupils  have  occasion 
to  write  them.  The  subject  of  numeration  is  here 
dwelt  upon  so  fully  because  it  is  so  intimately  con- 
nected with  the  decimal  system,  and  because  the 
decimal  system  is  usually  the  weakest  place  in  arith- 
metical teaching  in  this  country.  When  the  decimal 
system  of  numbers  and  the  Arabic  notation  are  thor- 
oughly understood,  arithmetic  is  half  learned.  Young 
teachers  are  therefore  earnestly  advised  to  advance 
.slowly  and  thoroughly  through  the  subject. 


ADDITION.  147 

37.   ADDITION. 

In  written  addition  it  is  more  convenient  to  begin 
with  the  lowest  place,  that  is,  with  the  units,  and  to 
work  towards  the  highest,  thus  reversing  the  process 
of  mental  addition.  The  first  examples  should  be  the 
addition  of  numbers  below  1,000,  because  the  pupils 
are  already  familiar  with  these  numbers. 

Attention  should  be  called  to  the  fact  that  2  boys 
and  3  slates  make  neither  5  boys  nor  5  slates,  etc. ; 
by  which  the  pupils  will  be  led  to  see  that  only  like 
quantities  can  be  added.  This  will  show  the  reason 
for  adding  units  to  units,  tens  to  tens,  hundreds  to 
hundreds,  etc.  ;  and  for  writing  numbers,  when  they 
are  to  be  added,  so  as  to  bring  units  under  units,  tens 
under  tens,  etc.,  as  a  matter  of  convenience.  These 
explanations  made,  introduce  an  example,  as  the  fol- 
lowing : 

3  First  add  the  column  of  units.     The  result 

88       is  32  units  ;  which  are  equal  to  3  tens  and  2 

9       units.      The   2    units    are   written   under  the 

45       units ;  and  the  3  tens  are  added  to  the  column 

13       of  tens.     The  result  is  23  tens,  equal  to  2  hun- 

77       dreds  and  3  tens.     The  whole    sum   is    thus 

232       found  to  be  two  hundreds,  three  tens,  and  two 

units,  or  232.    At  first  the  pupil  may  be  allowed 

to  write  the  tens   resulting  from  adding  the    units, 

with  a  small  figure  over  the  column  of  tens,  as  the 

3   above. 


148  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

After  the  pupils  have  had  considerable  practice  in 
adding  numbers  below  1,000,  the  addition  of  numbers 
above  1,000  may  be  introduced.     The  work  and  ex- 
planation will  appear  as  follows  : 
243i  The  column  of  units  give    1 5  units  —  i 

6742       ten  and  5  units.   Write  the  five  units  under 
982       the  line  in  the  column  of  units,  and  add  the 
3752       ten  to  the  column  of  tens.     This  gives  30 
97602       tens  =  3  hundreds  and  o  tens.     Write  the 
785       o  tens  under  the  tens,  and  add  the  3  hun- 
6742       dreds  to  the  column  of  hundreds.     There 
116,605      results   46   hundreds  -—  4  thousands  and  6 
hundreds.    Write  the  6  hundreds  under  the 
column  of  hundreds,  etc. 

The  above  examples  are  sufficiently  long  for  this 
stage  of  the  work.  Practical  problems  should  be  in- 
troduced constantly  ;  but  for  this  purpose  it  is  better 
to  depend  upon  a  good  text-book. 

Experience  shows  that  it  is  very  easy  to  make  mis- 
takes in  the  easiest  of  mathematical  processes,  that 
is,  in  addition  ;  so  that  when  certainty  of  results  is 
desired,  it  is  well  to  perform  the  additions  twice,  once 
beginning  at  the  bottom  of  the  columns  and  once  at 
the  top.  Or,  if  the  columns  are  very  long,  they  may 
be  divided  into  two  parts,  the  parts  added  separately, 
and  then  the  partial  results  added.  If  the  result  thus 
reached  agrees  with  the  result  of  adding  the  entire 
columns  at  once,  the  result  is  probably  right. 


SUBTRACTION.  149 

38.    SUBTRACTION. 

The  word  " subtraction"  means  taking  away.  In 
arithmetic  it  signifies  the  process  of  taking  one 
number  from  another,  or  of  finding  how  much  larger 
one  number  is  than  another,  that  is,  how  many  more 
units  one  number  contains  than  another.  The  num- 
ber which  is  to  be  diminished  is  called  the  minuend ; 
the  number  which  is  to  be  taken  away  is  called  the 
subtrahend  ;  the  number  which  is  left,  or  which  shows 
how  many  units  the  minuend  is  greater  than  the  sub- 
trahend, or  how  many  units  the  subtrahend  contains 
less  than  the  minuend,  is  called  the  remainder  or  dif- 
ference. 

These  definitions  should  be  developed  from  one  or 
two  examples  ;  as  3  from  5,  4  from  6.  It  would  not 
be  without  profit  to  illustrate  the  terms  by  perform- 
ing first  the  act  of  taking  one  number  of  objects 
from  another  number  ;  as,  for  example,  4  boys  from  6 
boys ;  and  then  the  act  of  comparing  one  number  with 
another,  to  find  the  difference ;  as,  for  example,  com- 
paring 4  boys  with  6  boys,  to  find  how  many  more 
there  were  in  one  group  than  in  the  other.  When- 
ever numbers  are  to  be  seen  in  new  relations,  the 
teacher  cannot  take  too  much  pains  to  make  sure 
that  the  ideas  of  the  numbers  are  clear  and  distinct. 

Write  the  minuend  under  the  subtrahend,  so  that, 
as  in  addition,  units  of  the  same  kind  will  stand  under 
one  another. 


150  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

Problems  should  be  given  first  in  which  each  figure 
in  the  subtrahend  stands  for  a  smaller  number  than 
the  corresponding  figure  in  the  minuend ;  as, 

976     5  units       from  6  units         =  I  unit ; 

—  435     3  tens  "     7  tens          =4  tens; 

==  541     4  hundreds  "      9  hundreds  =  5  hundreds. 

Therefore  there  remain  5  hundreds,  4  tens,  and  i 
unit  —  541. 

When  the  pupils  have  been  made  familiar  with 
such  examples,  problems  should  be  introduced  in 
which  one  or  more  figures  in  the  subtrahend  stand 
for  larger  numbers  than  the  corresponding  figures  in 
the  minuend.  The  two  following  examples  with  their 
explanations  will  make  the  principles  plain  upon  which 
they  are  to  be  solved. 

7  units  cannot  be  taken  from  5  units  ;  so 

495  we  separate  one  of  the  nine  tens  into  units, 
-257  which  gives  10  units;  and  these  10  units 
=  238  added  to  the  5  units  make  15  units.  From 
15  units  take  7  units,  and  8  units  remain. 
Then  8  tens,  which  were  left,  minus  5  tens,  leave  3 
tens;  and  4  hundreds  — 2  hundreds  =  2  hundreds. 
The  remainder,  then,  is  2  hundreds,  3  tens,  and  8 
units  =  238. 

The  following  problem  presents  an  additional  diffi- 
culty : 

99IO         2    units    cannot   be   taken   from   o   units. 

1000  Since,  now,  there  are  no  tens  and  no  hun- 
—  732  dreds,  we  change  I  thousand  into  10  hun- 
=  268  dreds ;  we  change  i  of  these  hundreds  into 


SUB  TRA  CTION.  1 5 1 

10  tens,  and  i  ten  into  10  units.  There  remain 
then  in  the  minuend  no  thousands,  but  9  hundreds,, 
9  tens,  and  10  units.  Then,  2  units  from  10  units  = 
8  units  ;  3  tens  from  9  tens  =  6  tens ;  and  7  hundreds, 
from  9  hundreds  -—  2  hundreds.  So  the  remainder  is 
2  hundreds,  6  tens,  8  units  =  268. 

Such  examples  as  the  last  two  may  be  readily 
solved  by  the  application  of  the  principle,  that  if  two 
numbers  are  equally  increased,  the  difference  remains 
the  same,  and  by  remembering  that  10  units  of  any 
order  is  equal  to  I  unit  of  the  next  higher  order : 

As  6  units  cannot  be  taken  from  4  units, 
274     add  10  units,  making  14  units ;  6  units  from 
-  146     14  units  leaves  8  units.     Add  i  ten  to  the  4 
—  128     tens,  making  5  tens,  which  taken  from  7  tens 
leaves  2  tens  ;  and  2  hundreds  —  i  hundred  = 
i   hundred.     So  that  the  remainder  is  i  hundred,  2 
tens,  8  units  =  128.    It  will  be  perceived  that  we  have 
added  10  units  to  the  minuend,  and  their  equivalent,. 
i   ten,  to  the  subtrahend ;   and,  consequently,  have 
not  changed  the  difference.    This  method  of  explana- 
tion and  practice  is  believed  to  be  easier  of  applica- 
tion than  the  method  first  explained,  and  is  therefore 
recommended. 

Since  the  minuend  —  the  subtrahend  =  the  remain- 
der, 

a.  The  minuend  —  the  remainder  =  subtrahend. 

b.  The  subtrahend  +  remainder  =  minuend. 


152  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

We  can  therefore  prove  the  correctness  of  the  work 
in  subtraction,  either  by  subtracting  the  remainder 
from  the  minuend,  in  which  case  the  subtrahend  is 
obtained,  or  by  adding  the  remainder  to  the  subtra- 
hend, in  which  case  we  obtain  the  minuend.  The 
latter  proof  is  more  practical  than  the  former,  and 
should  be  occasionally  used  by  the  pupils. 

All  the  principles  involved  in  subtraction  can  be 
learned  by  the  use  of  small  numbers  ;  so  that  it  is 
better  to  give  the  pupils  much  practice  with  these, 
before  introducing  large  numbers.  In  the  use  of 
practical  problems  small  numbers  are  much  prefera- 
ble ;  since  the  imagination  of  pupils  can  then  be 
more  easily  appealed  to  when  necessary. 

39.   MULTIPLICATION. 

The  meaning  of  the  terms  used  in  multiplication 
may  be  made  clear  to  the  pupils  in  the  following  way. 
Let  the  teacher  write,  in  a  horizontal  line  on  the 
board,  seven  dots,  and  ask,  "  How  many  dots  have  I 
made  ?  "  Then  let  him  make,  under  these,  two  rows 
more,  and  ask,  "  How  many  rows  of  seven  dots  each 
have  I  made  ?  How  many  times  are  seven  dots  re- 
peated ?  How  many  dots  are  there  in  all  ?  " 

Then  let  the  explanation  follow,  while  the  teacher 
continually  points  to  the  single  dots,  the  rows,  or  the 
whole  mass.  This  whole  process  is  called  multiplica- 
tion. The  number  of  dots  in  the  first  row,  namely  7, 


MULTIPLICATION.  153 

is  the  multiplicand.  The  number  of  rows,  namely  3, 
is  the  multiplier.  The  whole  number  of  dots,  namely 
21,  is  the  product.  We  have  repeated  7  3  times,  and 
the  result  is  21. 

The  work  on  the  board,  as  it  has  grown  up  under 
the  hand  of  the  teacher,  will  appear  thus  : 


When  the  pupils  have  followed  several  such  illus- 
trations, they  will  comprehend  the  following  defini- 
tions : 

Multiplication  is  the  process  of  finding  how  many 
units  result  from  repeating  a  number  a  given  number 
of  times.  The  number  to  be  repeated  is  called  the 
multiplicand.  The  multiplier  is  the  number  showing 
how  many  repetitions  are  to  be  made.  The  product 
is  the  number  showing  how  many  units  result  from 
the  repetitions.  The  multiplier  and  the  multiplicand 
are  called  the  factors  of  the  product.  Thus,  in  the 
above  example,  7  is  the  multiplicand,  3'  the  multiplier, 
21  the  product,  and  7  and  3  are  the  factors  of  21. 

It  is  not  worth  while  at  this  stage  to  have  these  defi- 
nitions committed  to  memory ;  but  the  terms  should 
be  thoroughly  understood,  so  that  they  will  bring  up, 
in  the  minds  of  the  pupils,  clear  and  distinct  ideas. 


154          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

a.  The  examples  first  introduced  for  explanation 
and  practice  should  contain  two  or  three  places  in 
the  multiplicand  and  only  one  in  the  multiplier.    The 
following  will  show  the  proper  explanation : 

3x7  units  =  21    units  =  2  tens  and   I    unit. 

247     Write  the   i   unit  under  the  column  of  units. 

3     3X4  tens  —  12  tens,  to  which  add  the  2  tens 

741     from  the  21  units,  and  the  sum  is   14  tens=  i 

hundred  and  4  tens.     Write  the  4  tens  under 

the  tens.    3x2  hundreds  =  6  hundreds,  to  which  add 

the   i   hundred  from  the   14  tens,  and  the  sum  is  7 

hundreds,  which  is  to  be  written  under  the  hundreds. 

The  result  is  7  hundreds,  4  tens,  and  i  unit  =  741. 

b.  Next  follow  examples  with  tens  only  in  the  mul- 
tiplier.     Here  should  come  the  explanation  of  the 
process  of  multiplying  a  number  by  10.     In  the  treat- 
ment of  numbers  from  i  to  1,000  it  was  shown  that, 

10  x  20  =  200,  10  x  89  =  890, 

10  x  27  =  270,  10  x  93  =  930, 

10  x  39  =  390,  10  x  72  =  720,  etc.  ; 

in  all  which  cases  we  obtained  just  as  many  tens  as 
there  were  units  in  the  multiplicand.  Now  since  tens 
are  indicated  by  a  zero  at  their  right,  to  multiply  a. 
number  by  10  we  have  only  to  put  a  zero  at  the  right, 
thus  setting  the  units'  figure  in  place  of  the  tens',  the 
tens'  figure  in  place  of  the  hundreds',  etc. 

Or  the  same  may  be  shown  by  such  examples  as, 
the  following : 


MUL  TIPLICA  TION.  I  5  5 

73     10X3  units  =  30  units  =  3  tens  ; 
X  10     10  X  7  tens  =  70  tens  =7  hundreds  ; 
730     and  7  hundreds  and  3  tens  —  730. 

Suppose,  now,  we  wish  to  multiply  a  number  by 
40.  We  may  first  multiply  by  4,  and  then  by  10, 
since  10x4  times  a  number  is  40  times  the  number. 
For  example,  40  x  23. 

23 

X40    4x23=92;  10x92  =  920. 
920 

It  follows  that  to  multiply  by  tens  we  have  only  to 
multiply  by  the  number  of  tens  and  put  a  cipher  after 
the  result. 

c.  The  third  class  of  problems  should  be  those  with 
tens  and  units  in  the  multiplier. 

4  X  38  units  ^152  units. 

38         38         20  X  38  =  2  x  10  X  38  =  760  =  76  tens  ; 

X  24     X  24     and  since  no  units  can  arise  from  the 

152       152     multiplication  of  any  number  by  tens, 

760      76       we  may  omit  the  zero  as  in  the  second 

912      912     example   in    the   margin,  and  begin  to 

write  the  product  under  the  tens. 
•Now,  since  24  times  a  number  is  4  times  the  num- 
ber plus  20  times  the  number,  we  have  only  to  add  the 
partial  products  in  order  to  obtain  the  entire  product. 

d.  Finally,  examples    should   be   introduced  with 
three  or  more  figures  in  the   multiplier ;   as,  243  x 
35/6. 


156  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

This  is  equivalent  to  saying  multiply  3,576  by  3,  by 
40,  and  by  200,  and  add  the  results. 

3  X  3576=  10728,  to  be  written  units  un- 
3576     der  units,  etc. 

243         40  X  3576  =  143040  =  14304  tens,  to  be 
10728     written  tens  under  tens,  etc. 
14304          200  x  3576  =  2  x  100  x  3576  =  2  x  10  x  10 
7152         X  3576  =  715200  =  7152    hundreds,   to    be 


868,968     written  hundreds  under  hundreds,  etc. 

If,  now,  we  add  3  times,  40  times,  and  200  times 
the  number  together,  we  have  243  times  the  number. 
If  the  pupil  has  observed  that  multiplying  numbers 
by  10,   100,    1,000,  etc.,  simply  sets  the  figures  one, 
two,  three,  etc.,  places  towards  the  left,  he  at  once 
comprehends  the  reason  for  the  rule :  Write  the  first 
figure  of  each  partial  product  under  the  place  of  the 
number  with  which  you  multiply.     If  you  multiply 
by  units,      write  the  first  figure  under  units  ; 
"   tens,  "       "      "         "          "     tens; 

"   hundreds,  "       "      "         "          "      hundreds; 
etc.  etc. 

e.  If  the  pupil  thoroughly  comprehends  the  in- 
struction above  suggested,  he  will  have  little  diffi- 
culty with  numbers  containing  ciphers, 

20 1 

X4Q3 
603 
804 

81,003 


DIVISION.  157 

The  number  201  is  to  be  repeated  3  times;  then 
400  (loox  4)  times  ;  and  then  the  partial  products  are 
to  be  united. 


x  30080 
6724800 
252180 


2,528,524,800 

The  number  is  to  be  repeated,  first,  80  (10  X  8) 
times,  then  30,000  (10,000  X  3)  times,  and  the  partial 
products  added. 

Such  examples  as  the  last  would  be  introduced  at 
this  stage  only  for  the  benefit  of  the  brightest  pupils, 
who,  by  such  work,  may  be  interested  and  benefited. 

Perhaps  the  easiest  proof  of  multiplication  is  to 
make  the  multiplier  a  multiplicand  and  the  multipli- 
cand a  multiplier,  and  multiply  again.  Later,  the 
product  may  be  made  a  dividend  and  the  multiplicand  a 
divisor,  when  the  quotient  should  equal  the  multiplier. 

Practical  problems  are  to  be  introduced  at  each 
stage  of  the  work  of  multiplication,  which  is  heyre 
marked  a,  b,  c,  dy  and  e.  For  most  of  these,  however, 
the  teacher  should  depend  upon  a  good  text-book ; 
and  this  should  be,  a  part  of  the  time,  in  the  hands  of 
the  pupils. 

4O.    DIVISION. 

The  number  to  be  divided  is  called  the  dividend ; 
the  number  by  which  we  divide,  the  divisor ;  the 
number  which  shows  how  many  times  the  divisor  is 


158  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

contained  in  the  dividend,  or  what  part  of  the  divi- 
dend the  divisor  is,  the  quotient ;  the  number  left 
when  the  division  is  not  completed,  the  remainder. 
For  example,  the  process  of  finding  how  many  times 
7  is  contained  in  21,  or  what  a  seventh  part  of  21  is, 
is  the  division  of  21  by  7.  Here  21  is  the  dividend, 
7  is  the  divisor,  and  3  the  quotient.  Had  we  at- 
tempted to  find  how  many  times  7  is  contained  in  23, 
we  should  have  found  that  it  was  contained  3  times, 
with  a  remainder  of  2. 

Whether  we  attempt  to  find  how  many  times  a 
number  is  contained  in  a  given  number,  or  what  the 
corresponding  part  of  the  given  number  is,  the  pro- 
cess is  the  same ;  for  example,  take  the  above  num- 
bers, 21  and  7.  The  seventh  part  of  7  is  i,  the 
.seventh  part  of  2  times  7  is  2  ;  and,  in  general,  the 
seventh  part  of  21  is  as  many  units  as  7  is  contained 
times  in  21,  namely  3.  So  that  it  is  not  necessary  to 
consider  the  two  kinds  of  division  separately,  although 
the  pupil  should  always  be  required  to  know  and  to 
state  what  he  is  doing. 

The  degree  of  difficulty  in  division  depends  upon 
the  constitution  of  the  divisor ;  so  that  the  divisor 
determines  the  stages  of  the  pupils'  work  in  division. 
They  are  the  following  : 

a.  The  divisor  is  composed  of  units ; 

b.  "         "       "          "          "  tens ; 

c.  "          "       "          "          "  tens  and  units ; 

d.  "          "  contains  2,  3,  etc.,  places. 


DIVISION.  1 59 

Since,  however,  it  is  easier  to  make  the  process 
understood  if  the  dividend  is  small,  examples  should 
be  chosen  for  the  first  work  where  that  is  the  case. 

DIVIDING  BY   UNITS. 

We  will  first  explain  an  example  of  division  by  2. 

a  b                                     c 

2)759(379  2)759(300  £  of  759 

6 . .  x  2  600  \  "  600  ==  300 

15.  758  iS9(  70  £  "  140-   70 

14         i  140  £  "     18=     9 

19  759  ~i9(__9  i  "       i  =       i 

18  ^8379  iof759=379£ 

i  i 

#.  The  number  759  consists  of  7  hundreds,  5  tens, 
and  9  units.  We  first  divide  6  hundreds  by  2,  and 
we  have  3  hundreds.  These  3  hundreds  we  write  at 
the  right.  We  indicate  that  2X3  hundreds,  or  6 
hundreds,  have  been  divided  by  writing  6  under  7. 
We  subtract  6  hundreds  from  7  hundreds,  and  i  hun- 
dred remains,  to  which  we  unite  the  5  tens,  and  have 
15  tens.  We  divide  14  tens  by  2,  and  the  result  is  7 
tens,  which  we  set  in  the  tens'  place.  We  write  2x7 
tens=  14  tens  under  the  tens  to  show  that  they  have 
been  divided.  We  subtract  the  14  tens,  and  there  re- 
mains i  ten,  to  which  we  add  the  9  units,  and  we 
have  19  units.  We  divide  18  units  by  2,  and  obtain 
9  units,  which  we  write  in  the  units'  place  in  the  quo- 


1 60          ARITHMETIC  IN  PRIMARY  SCHOOLS. 

tient.  We  subtract  2x9  units,  or  18  units,  from  the 
19  units,  and  have  a  remainder  of  i.  So  that  the  re- 
sult, or  product,  is  371,  and  I  remainder. 

Under  b  the  division  is  indicated  more  fully.  The 
parts  of  the  dividend  which  have  been  divided  (600, 
140,  1 8),  as  well  as  the  parts  of  the  quotient  (300,  70, 
and  9),  are  written  out  in  full.  The  third  form,  c,  is 
the  form  with  which  the  pupil  is  familiar  in  his  men- 
tal work.  It  is  added  here  to  make  the  new  and 
shortened  form  of  division  still  clearer.  The  form  b 
is  recommended  only  for  the  purpose  of  explaining 
the  reason  of  form  c.  The  first  form  is  the  one  to  be 
used  in  practical  work. 

Here,  more  than  anywhere  else,  is  it  necessary  for 
the  pupil  to  write  the  different  figures  in  the  proper 
places,  —  units  under  units,  tens  under  tens,  etc.  To 
assist  the  teacher  in  securing  this,  the  division  of  the 
board,  and  also  the  slate,  into  little  rectangles,  as 
formerly  advised,  is  very  helpful. 

Abundant  practice  in  dividing  by  2  should  precede 
the  dividing  by  other  units.  A  clear  comprehension 
of  the  reason  for  the  different  parts  of  the  process,  as 
well  as  great  facility  in  the  operation,  cannot  be  too 
strenuously  insisted  upon,  before  the  pupil  is  allowed 
to  go  on  to  new  work.  Time  spent  here  is  more  than 
saved  later. 

It  is  important  that  the  pupil  learn  to  determine, 
as  soon  as  he  begins  to  divide,  how  many  places  there 
must  be  in  the  quotient ;  because  this  explains  the 


DIVISION.  l6l 

reason  for  putting  a  zero  in  the  quotient,  whenever 
the,  divisor  is  not  contained  in  the  number  of  units  of 
any  order  in  the  multiplicand.  An  example  will  make 
this  clear. 

Since    I    hundred-thousand    can- 
6)184549(30,758     not    be  divided  by  6  and  produce 
1 8  a  whole  number,  we  divide  18  ten- 

45  thousands  by  6,  and  the  result  is  3 

42  ten-thousands.    This  shows  the  pu- 

34  pil  that  there  must  be  5  places  in 

30  the  quotient,  which    the    beginner 

49  may  indicate  by  5  points.     Since  4 

48  thousands  divided  by  6  produce  no 

i  thousands,  a  zero  must  be  put  in 

the  quotient  in  the  thousands'  place ; 
else  the  quotient  would  not  contain  5  places,  and  the 
first  figure,  3,  would  be  read  3  thousands  ;  for  4  thou- 
sands and  5  hundreds,  or  45  hundreds,  divided  by  6, 
give  7  hundreds.  By  such  examples  the  pupil  will 
learn  to  put  a  zero  in  the  quotient  whenever  the  num- 
ber shown  by  bringing  down  a  figure  of  the  dividend 
is  not  divisible  by  the  divisor. 

The  correctness  of  the  work  in  division  may  be 
tested  by  multiplying  the  divisor  by  the  quotient,  and 
adding  the  remainder  to  the  product.  The  sum  should 
equal  the  dividend. 

After  the  pupils  have  had  a  good  deal  of  practice 
in  dividing  by  numbers  represented  by  one  figure, 
using  the  form  given  above,  they  may  be  allowed  to 


1 62  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

divide    by    the    same    numbers,    writing .  simply   the 
divisor  under  the  dividend,  thus  : 

2)9151712  6  i 
478630—1  remainder. 

At  first  the  remainder  may  be  written  in  small  fig- 
ures over  and  a  little  at  the  left  of  the  next  place. 

DIVIDING  BY   TENS. 

It  is  well  to  make  a  distinct  step  of  dividing  by 
numbers  consisting  of  tens  only,  because  it  throws 
light  on  the  succeeding  steps  in  division.  It  would 
be  profitable,  at  this  point,  to  review  the  mental  pro- 
cesses of  multiplying,  and  also  of  dividing,  by  10,  20, 
30,  etc.,  to  100.  This  done,  an  example  or  two  will 
make  this  step  understood. 

The  twentieth  of  41   hundreds  is  2 

20)4165(208     hundreds,   and  a  remainder  of   I   hun- 

40  dred ;    to  this    I    hundred,  or   10  tens, 

165  add  6  tens,  and  the  sum  is   16  tens ; 

1 60  which  is   not  divisible   by  20,  and    so 

5  there  are  no  tens  in  the  quotient,  and 

the  tens'  place  must  be  filled  with    a 

zero.     Dividing  165  units  by  20,  and  we  have  8  units, 

with  a  remainder  of  5- 

It  will  soon  be  obvious  to  the  pupils  that  dividing 
by  10  is  accomplished  by  cutting  off  the  unit  figure, 
and  regarding  it  as  representing  tenths ;  and  so, 
later,  of  dividing  by  100,  1,000,  etc. 


DIVISION.  165 

DIVIDING  BY  NUMBERS   OF  TWO   PLACES. 

The  difficulty  of  dividing  by  such  numbers  arises 
from  the  fact  that  the  pupils  do  not  know  the  multi- 
plication table  for  numbers  so  large  ;  and  hence  the 
products  of  these  numbers,  that  is,  the  divisors  used,, 
must  be  found.  For  this  purpose  it  is  often  neces- 
sary for  the  pupils  at  first  to  proceed  by  way  of  trial. 
This  trial  consists  in  finding  how  often  a  convenient 
number  of  about  the  same  size  as  the  divisor  —  that 
is,  a  number  whose  product  by  2,  3,  4,  etc.,  to  9,  is 
already  known  —  is  contained  in  the  number  to  be 
divided,  and  then  multiplying  the  divisor  by  this  quo- 
tient. For  example,  if  I  wish  to  find  how  many 
times  53  is  contained  in  480,  I  first  see  how  often  50 
is  contained  in  it.  This  I  know  by  knowing  the 
products  of  the  tens,  that  is,  20,  30,  etc.,  by  2,  3,  etc. 
Since  50  is  contained  9  times  in  480,  therefore  it  is 
probable  that  S3  is  contained  9  times;  this  is  here 
the  fact,  for  9  X  53  =477.  Here  the  probability  and 
the  truth  agree  ;  but  were  the  divisor  54,  this  would 
not  be  the  case  ;  for  9  x  54  —  486.  In  this  case  the 
quotient  must  be  diminished  by  i.  The  nearer  the 
convenient  number,  or  trial  divisor,  is  to  the  true 
divisor,  the  greater  the  probability  is  that  the  trial 
quotient  will  prove  to  be  the  true  quotient.  Hence 
in  dividing  by  56,  57,  58,  or  59,  it  would  be  better  to 
use  60,  rather  than  50,  as  a  trial  divisor,  while  50 
would  be  more  likely  than  60  to  give  us  the  true  num- 
ber, if  we  were  dividing  by  50,  51,  52,  53,  or  54.  In 


1  64  ARITHMETIC  IN  PRIMARY  SCHOOLS. 

the  former  case  it  would  often  be  necessary  to  in* 
crease  the  trial  quotient  by  I  in  order  to  obtain  the 
true  quotient. 

After  much  experience,  and  practice,  the  pupil  can 
make  this  trial  in  his  mind  without  writing  down  any 
of  the  work  ;  and  this  poVer  the  teacher  should  try 
to  develop  in  the  pupil. 

The  following  is  a  typical  explanation  :  7  hundreds 

-5-23  gives  no  hundreds;  7  hundreds  +  5 

23)75I(32     tens  ^75  tens;  75  tens  -5-23  =  3  tens,  for 

69  3x23  tens  =  69  tens.     There    remain  6 

6  1  tens,  to  which  add   I  unit,  and  we  have 

46          6  1  units.     6  1   units  —  20  =  3;  but  3x23 

15  =69  ;  so  23  is  not  contained  in  61   units 

3  times,  but  one  less  than  3  times,  or  2 

times  ;  2  X  23  =46  ;  and  61  —  46=  15,  the  remainder. 


8497 

74Q32  X9254 

46005  33988 

37016          42485 

89899         l6994 
83286        76473 


66134  78631238 

64778  1356 

I3S6  78632594 

7  ten-millions  are  not  divisible  by  9254; 
78  millions  are  not  divisible  by  9254 ; 
786  hundred-thousands  are  not  divisible  by  9254; 


DIVISION.  165 

7863  ten-thousands  are  not  divisible  by  9254; 
78632  thousands  -f-  9254  =  8000  ;  8000  X  9254  = 
74032  thousands,  which  taken  from  78632  thousands 
leave  4600  thousands  =  46000  hundreds,  to  which 
add  5  hundreds,  and  we  have  46005  hundreds  ;  46005 
hundreds -^92  54  =  400;  400  x  9254  =  37016  hundreds, 
which  subtracted  from  46005  hundreds  leaves  8989 
hundreds  —  89890  tens,  to  which  add  9  tens,  and  the 
sum  is  89899  tens;  89899  tens  -f-  9254  =  90  ;  90  X 
9254=83286  tens,  which  taken  from  89899  tens 
leave  6613  tens  =  66130  units,  to  which  add  4  units, 
and  we  have  66134  units  ;  66134  units  -^9254  =  7  ;  7 
X  9254=  64778,  which  from  66134  =  1356,  remainder. 

Of  course  examples  of  this  length  would  be  given 
only  to  older  and  brighter  pupils. 

The  pupil  can  now  prove  his  multiplication  by 
dividing  the  product  either  by  the  multiplier  or  the 
quotient.  If  the  work  is  right,  the  quotient  will  be 
the  other  factor. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


REC'D  LD 

JUL  1 8  1962 


50rn-7,'16 


